{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":59671,"title":"Write a code to implement Euler's method to integrate a simple function","description":"Euler's method approximates the solution to a differential equation as\r\n\r\nwhere . The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over  equally-spaced points ( equal intervals) between times  and . You must implement the boundary condition . 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style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.992px 8px; transform-origin: 213.992px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEuler's method approximates the solution to a differential equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.9167px 8px; transform-origin: 73.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally-spaced points (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39.5\" height=\"18\" style=\"width: 39.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.4167px 8px; transform-origin: 98.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equal intervals) between times \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.767px 8px; transform-origin: 111.767px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. You must implement the boundary condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18.5\" style=\"width: 63px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t,x] = EulerIntegration(t0,tf,N,x0,func)\r\n  t = 0;\r\n  x = 0;\r\nend","test_suite":"%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) 2*x.*t;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/N;\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    x(i+1) = x(i) + h*func(t(i),x(i));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":4078801,"edited_by":223089,"edited_at":"2025-10-10T13:57:39.000Z","deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":"2025-10-10T13:57:39.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-02-28T23:15:52.000Z","updated_at":"2026-03-23T10:41:43.000Z","published_at":"2024-02-28T23:15:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuler's method approximates the solution to a differential equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + h \\\\cdot f(x, t )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh = \\\\Delta t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equally-spaced points (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equal intervals) between times \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_o\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. You must implement the boundary condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(0)=x0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60983,"title":"Check Euler's characteristic on regular polyhedra","description":"Problem statement\r\nGiven the number of vertices and the number of faces of a given regular polyhedron, compute the number of its edges with Euler characteristic, a powerful generic formula linking these three.\r\n\r\nExamples -\u003e check the test suite\r\n\r\nForbidden functions / expressions\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nMesh processing\r\nMesh generation toolbox","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 414.733px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 207.367px; transform-origin: 408px 207.367px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 263.333px 8px; transform-origin: 263.333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the number of vertices and the number of faces of a given regular polyhedron, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.042px 8px; transform-origin: 110.042px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ecompute the number of its edges\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with Euler characteristic, a powerful generic formula linking these three.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.425px 8px; transform-origin: 6.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.8917px 8px; transform-origin: 59.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echeck the test suite\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/57483\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh processing\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/85173-mesh-generation-toolbox?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh generation toolbox\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function e = check_Euler_chracteristic(v,f)\r\n  e = f;\r\nend","test_suite":"%% tetrahedron\r\nv = 4;\r\nf = 4;\r\ne_correct = 6;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% octahedron\r\nv = 6;\r\nf = 8;\r\ne_correct = 12;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% hexahedron / cube\r\nv = 8;\r\nf = 6;\r\ne_correct = 12;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% icosahedron\r\nv = 12;\r\nf = 20;\r\ne_correct = 30;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% dodecahedron\r\nv = 20;\r\nf = 12;\r\ne_correct = 30;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Euler_chracteristic.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:45:16.000Z","deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-23T20:08:07.000Z","updated_at":"2026-02-10T17:11:09.000Z","published_at":"2025-07-24T07:25:21.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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It's possible to improve on Euler's method by doing the following:\r\n\r\n\r\nThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over  equally-spaced points ( equal intervals) between times  and . You must implement the boundary condition . 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10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEuler's method approximates the solution to a differential equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"177\" height=\"18.5\" style=\"width: 177px; height: 18.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"18\" style=\"width: 44.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. It's possible to improve on Euler's method by doing the following:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"133.5\" height=\"19.5\" style=\"width: 133.5px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 34.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 17.3333px; text-align: left; transform-origin: 384.5px 17.3333px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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width=\"283.5\" height=\"34.5\" style=\"width: 283.5px; height: 34.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.9167px; text-align: left; transform-origin: 384.5px 31.9167px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equally-spaced points (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39.5\" height=\"18\" style=\"width: 39.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equal intervals) between times \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. You must implement the boundary condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18.5\" style=\"width: 63px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t,x] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n  t = 0;\r\n  x = 0;\r\nend","test_suite":"%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) 2*x*t;\r\nf=func;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/(N-1);\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    xstar = x(i) + h*f(t(i),x(i));\r\n    x(i+1) = x(i) + h/2*(f(t(i),x(i)) + f(t(i+1),xstar));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n\r\n%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) x+t;\r\nf=func;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/(N-1);\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    xstar = x(i) + h*f(t(i),x(i));\r\n    x(i+1) = x(i) + h/2*(f(t(i),x(i)) + f(t(i+1),xstar));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4078801,"edited_by":4078801,"edited_at":"2024-03-05T14:42:54.000Z","deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2024-03-05T14:42:54.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-02-29T19:23:54.000Z","updated_at":"2026-03-31T11:25:40.000Z","published_at":"2024-02-29T19:23:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuler's method approximates the solution to a differential equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + h \\\\cdot f(x, t )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh = \\\\Delta t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It's possible to improve on Euler's method by doing the following:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^* = x(t) + h \\\\cdot f(x,t)\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + \\\\frac{h}{2} \\\\cdot (f(x,t) + f(t+\\\\Delta t, x^* ) )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equally-spaced points (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equal intervals) between times \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_o\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. You must implement the boundary condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(0)=x0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60942,"title":"Euler primes","description":"Historical context\r\n\r\n\r\nThe great swiss mathematician Leonhard Euler found the formula  to give some prime numbers.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 41, list the Euler prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 41, your function should return ep = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 41 then ep = 41 and f = 1 since 41 = P(0) and it is the only prime number m greater or equal and less or equal to 41;\r\nIf m = 100 then ep = [41, 43, 47, 53, 61, 71, 83, 97] and f = 8/13 since 41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), and in total there are 13 prime numbers in the range [41; 100] ; \r\nIf m = 19 then ep = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\nn is less or equal to 107 here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\n\r\nSee also\r\nProblem 60944. Ulam primes first formula, Problem 60943. Ulam primes second formula\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 988.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 494.233px; transform-origin: 408px 494.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; 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\"\u003e\u003cspan style=\"\"\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"125\" height=\"19.5\" style=\"width: 125px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euler prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 286.658px 8px; transform-origin: 286.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.9833px 8px; transform-origin: 28.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e41 = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.1167px 8px; transform-origin: 10.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 110.258px 8px; transform-origin: 110.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 8/13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 141.192px 8px; transform-origin: 141.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand in total there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [41; 100] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 19 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3417px 8px; transform-origin: 58.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e107\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60944. Ulam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60943. Ulam primes second formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euler_primes(m)\r\n  ep = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 41;\r\nep_correct = 41;\r\nf_correct = 1;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97];\r\nf_correct = 8/13;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197];\r\nf_correct = 13/34;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 19;\r\nep_correct = [];\r\nf_correct = 0;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euler_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:49:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2025-07-09T05:56:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T14:44:30.000Z","updated_at":"2026-03-16T19:16:06.000Z","published_at":"2025-06-26T15:44:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = n^2 + n + 41}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euler prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 8/13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand in total there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [41; 100] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 19 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e107\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60944. Ulam primes first formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60943. Ulam primes second formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44668,"title":"Day counter function","description":"Write a function called day_counter that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\r\nYou are not allowed to use the datenum built-in function.\r\n(Inspired by Project Euler.)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 144px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 72px; transform-origin: 407px 72px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70px 8px; transform-origin: 70px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function called\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eday_counter\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 265px 8px; transform-origin: 265px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 199px 8px; transform-origin: 199px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYou are not allowed to use the datenum built-in function.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93px 8px; transform-origin: 93px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e(Inspired by Project Euler.)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"% function to count number of mondays\r\nfunction foms = day_counter( year )\r\n\r\nend","test_suite":"%%\r\nyear  = 1902;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2002;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 1996;\r\nfoms_correct = 3;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 1885;\r\nfoms_correct = 1;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2000;\r\nfoms_correct = 1;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2013;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":171559,"edited_by":223089,"edited_at":"2023-01-16T11:22:13.000Z","deleted_by":null,"deleted_at":null,"solvers_count":77,"test_suite_updated_at":"2023-01-16T11:22:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-05-30T02:42:41.000Z","updated_at":"2026-01-01T20:23:31.000Z","published_at":"2018-05-30T02:42:41.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function called\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eday_counter\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYou are not allowed to use the datenum built-in function.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(Inspired by Project Euler.)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42821,"title":"Polygon division","description":"Given the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\r\n\r\n1. The division is done by drawing straight lines between existing vertices.\r\n\r\n2. The triangles are made of existing vertices.\r\n\r\n3. Different orientations of a similar solution are counted as different solutions.\r\n\r\nAssume that n is a positive integer greater than 2.\r\n\r\nExample 1:\r\n\r\nn = 4 (square)\r\n\r\nw = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\r\n\r\nExample 2:\r\n\r\nn = 5 (pentagon)\r\n\r\nw = 5","description_html":"\u003cp\u003eGiven the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\u003c/p\u003e\u003cp\u003e1. The division is done by drawing straight lines between existing vertices.\u003c/p\u003e\u003cp\u003e2. The triangles are made of existing vertices.\u003c/p\u003e\u003cp\u003e3. Different orientations of a similar solution are counted as different solutions.\u003c/p\u003e\u003cp\u003eAssume that n is a positive integer greater than 2.\u003c/p\u003e\u003cp\u003eExample 1:\u003c/p\u003e\u003cp\u003en = 4 (square)\u003c/p\u003e\u003cp\u003ew = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\u003c/p\u003e\u003cp\u003eExample 2:\u003c/p\u003e\u003cp\u003en = 5 (pentagon)\u003c/p\u003e\u003cp\u003ew = 5\u003c/p\u003e","function_template":"function w = polydiv(n)\r\n  w = n;\r\nend","test_suite":"%%\r\nfiletext = fileread('polydiv.m');\r\nassert(isempty(strfind(filetext,'str')))\r\n\r\n%%\r\nn = 3;\r\nw_correct = 1;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 4;\r\nw_correct = 2;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 5;\r\nw_correct = 5;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 8;\r\nw_correct = 132;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 11;\r\nw_correct = 4862;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 15;\r\nw_correct = 742900;\r\nassert(isequal(polydiv(n),w_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":15521,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2016-04-30T20:32:42.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2016-04-24T15:28:47.000Z","updated_at":"2026-01-19T17:32:36.000Z","published_at":"2016-04-24T15:28:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e1. The division is done by drawing straight lines between existing vertices.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e2. The triangles are made of existing vertices.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e3. Different orientations of a similar solution are counted as different solutions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that n is a positive integer greater than 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 4 (square)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ew = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample 2:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 5 (pentagon)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ew = 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1483,"title":"Number of paths on a grid","description":"\r\nConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. \r\nYour destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary. \r\n\r\nEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.   \r\n\r\n4x3 has 10 ways\r\n\r\n6x5 has 126 ways\r\n\r\nThis problem can be solved using dynamic programming but there are other methods too. \r\n\r\nProblem 7)\r\nPrev: \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1482 1482\u003e\r\nNext: \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1484 1484\u003e","description_html":"\u003cdiv style = \"text-align: start; line-height: 20px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; transform-origin: 332px 148.5px; vertical-align: baseline; perspective-origin: 332px 148.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; perspective-origin: 309px 63px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. Your destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 21px; white-space: pre-wrap; perspective-origin: 309px 21px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e4x3 has 10 ways\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e6x5 has 126 ways\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis problem can be solved using dynamic programming but there are other methods too.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProblem 7) Prev:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www.mathworks.com/matlabcentral/cody/problems/1482\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e1482\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e Next:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www.mathworks.com/matlabcentral/cody/problems/1484\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e1484\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = paths2dest_ongrid(n,m)\r\n  y = n+m;\r\nend","test_suite":"%%\r\nm = 1; n = 1 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 2; n = 2 ;\r\ny_correct = 2;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 4; n = 3 ;\r\ny_correct = 10;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 6; n = 5 ;\r\ny_correct = 126;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 5; n = 5 ;\r\ny_correct = 70;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 1; n = 100 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 100; n = 1 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 2; n = 100 ;\r\ny_correct = 100;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 100; n = 2 ;\r\ny_correct = 100;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 15; n = 20 ;\r\ny_correct = 818809200;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":5,"created_by":11275,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":81,"test_suite_updated_at":"2020-09-28T20:02:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-05-01T14:58:23.000Z","updated_at":"2026-03-19T08:10:31.000Z","published_at":"2013-05-01T14:58:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. Your destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4x3 has 10 ways\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6x5 has 126 ways\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem can be solved using dynamic programming but there are other methods too.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 7) Prev:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1482\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e1482\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e Next:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1484\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e1484\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42938,"title":"Project Euler: Problem 18, Maximum path sum I","description":"By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\r\n\r\n       3*\r\n     7*  4\r\n   2   4*  6\r\n 8   5   9*  3\r\n \r\n3 + 7 + 4 + 9 = 23\r\n\r\nFind the maximum total from top to bottom of a given triangle.\r\n","description_html":"\u003cp\u003eBy starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\u003c/p\u003e\u003cpre\u003e       3*\r\n     7*  4\r\n   2   4*  6\r\n 8   5   9*  3\u003c/pre\u003e\u003cp\u003e3 + 7 + 4 + 9 = 23\u003c/p\u003e\u003cp\u003eFind the maximum total from top to bottom of a given triangle.\u003c/p\u003e","function_template":"function s = maxPathSum(tr)\r\n    % tr: lower triangular matrix\r\n  s = inf;\r\nend","test_suite":"%%\r\nx = [3 0 0 0\r\n  7 4 0 0\r\n  2 4 6 0\r\n  8 5 9 3];\r\nassert(isequal(maxPathSum(x),23))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(magic(10))),891))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(magic(100)^2)),251716502500))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(mod(magic(1000),997))),741570))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(gallery('integerdata',1e4,5e3,4))),37455089))","published":true,"deleted":false,"likes_count":4,"comments_count":2,"created_by":85274,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":144,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-08-30T10:52:35.000Z","updated_at":"2026-01-05T00:26:41.000Z","published_at":"2016-08-30T10:53:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBy starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[       3*\\n     7*  4\\n   2   4*  6\\n 8   5   9*  3]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e3 + 7 + 4 + 9 = 23\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the maximum total from top to bottom of a given triangle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45429,"title":"Prime Sum","description":"What is the minimum value that can be written as sum of primes in n different ways?\r\n\r\nFor example,\r\n 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\r\n\r\n \r\n 2 + 2 + 2 + 2 + 2\r\n 2 + 2 + 3 + 3\r\n 2 + 3 + 5\r\n 5 + 5\r\n 3 +7\r\n\r\n\r\n8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\r\n \r\n\r\n\u003chttps://projecteuler.net/problem=77\u003e","description_html":"\u003cp\u003eWhat is the minimum value that can be written as sum of primes in n different ways?\u003c/p\u003e\u003cp\u003eFor example,\r\n 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\u003c/p\u003e\u003cpre\u003e 2 + 2 + 2 + 2 + 2\r\n 2 + 2 + 3 + 3\r\n 2 + 3 + 5\r\n 5 + 5\r\n 3 +7\u003c/pre\u003e\u003cp\u003e8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\u003c/p\u003e\u003cp\u003e\u003ca href = \"https://projecteuler.net/problem=77\"\u003ehttps://projecteuler.net/problem=77\u003c/a\u003e\u003c/p\u003e","function_template":"function i = prime_sum(n)","test_suite":"%%\r\nassert(isequal(prime_sum(3),7))\r\n%%\r\nassert(isequal(prime_sum(6),11))\r\n\r\n%%\r\nassert(isequal(prime_sum(500),45))\r\n\r\n%%\r\nassert(isequal(prime_sum(2678),64))\r\n\r\n%%\r\nassert(isequal(prime_sum(4290),70))\r\n\r\n%%\r\nassert(isequal(prime_sum(6500),75))\r\n\r\n%%\r\nassert(isequal(prime_sum(116500),117))\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":363598,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-04-09T04:25:11.000Z","updated_at":"2020-04-09T04:25:11.000Z","published_at":"2020-04-09T04:25:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the minimum value that can be written as sum of primes in n different ways?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 2 + 2 + 2 + 2 + 2\\n 2 + 2 + 3 + 3\\n 2 + 3 + 5\\n 5 + 5\\n 3 +7]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://projecteuler.net/problem=77\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://projecteuler.net/problem=77\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44784,"title":"Project Euler: Problem 14 Longest Collatz sequence(harder version)","description":"This problem is a harder version of \"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\", because of time limits. \u003chttps://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003e \r\n\r\nThe following iterative sequence is defined for the set of positive integers:\r\nn → n/2 (n is even) n → 3n + 1 (n is odd)\r\nUsing the rule above and starting with 13, we generate the following sequence:\r\n13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.\r\nWhich starting number, no more than N, produces the longest chain, and how long? Don't cheat!","description_html":"\u003cp\u003eThis problem is a harder version of \"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\", because of time limits. \u003ca href = \"https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\"\u003ehttps://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003c/a\u003e\u003c/p\u003e\u003cp\u003eThe following iterative sequence is defined for the set of positive integers:\r\nn → n/2 (n is even) n → 3n + 1 (n is odd)\r\nUsing the rule above and starting with 13, we generate the following sequence:\r\n13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.\r\nWhich starting number, no more than N, produces the longest chain, and how long? Don't cheat!\u003c/p\u003e","function_template":"function y = euler014_v2(x)\r\n  y = x;\r\nend","test_suite":"1\r\n%%\r\nassessFunctionAbsence({'tic','toc','pause','etime','clock','now','str2num','timer'},'FileName','euler014_v2.m')\r\n\r\n2\r\n%%\r\nN = 2e8;\r\nnum_correct = 169941673;\r\nlen_correct = 954;\r\ntic\r\n[num, len] = euler014_v2(N);\r\nt=toc\r\nassert(t\u003e1);\r\nassert(isequal(num, num_correct));\r\nassert(isequal(len, len_correct));\r\n\r\n3\r\n%%\r\nN = 4e8;\r\nnum_correct = 268549803;\r\nlen_correct = 965;\r\ntic\r\n[num, len] = euler014_v2(N);\r\nt=toc\r\nassert(t\u003e1);\r\nassert(isequal(num, num_correct));\r\nassert(isequal(len, len_correct));\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":3668,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":"2018-11-11T06:11:05.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-11-09T16:59:43.000Z","updated_at":"2018-11-11T06:11:05.000Z","published_at":"2018-11-09T16:59:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem is a harder version of \\\"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\\\", because of time limits.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe following iterative sequence is defined for the set of positive integers: n → n/2 (n is even) n → 3n + 1 (n is odd) Using the rule above and starting with 13, we generate the following sequence: 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1. Which starting number, no more than N, produces the longest chain, and how long? Don't cheat!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":59671,"title":"Write a code to implement Euler's method to integrate a simple function","description":"Euler's method approximates the solution to a differential equation as\r\n\r\nwhere . The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over  equally-spaced points ( equal intervals) between times  and . You must implement the boundary condition . ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 124px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 62px; transform-origin: 408px 62px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.992px 8px; transform-origin: 213.992px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEuler's method approximates the solution to a differential equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"177\" height=\"18.5\" style=\"width: 177px; height: 18.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 32px; text-align: left; transform-origin: 385px 32px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"18\" style=\"width: 44.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 322.2px 8px; transform-origin: 322.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.9167px 8px; transform-origin: 73.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally-spaced points (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39.5\" height=\"18\" style=\"width: 39.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.4167px 8px; transform-origin: 98.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equal intervals) between times \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.767px 8px; transform-origin: 111.767px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. You must implement the boundary condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH4AAAAlCAYAAACEal28AAAFpklEQVR4Xu2bPagcVRTH3+sVUauIRIgWCoFYqBGCgkIURBEJJIrNq9QUIZUhSSUWEZIqpIhaioUKFiEgaJFCSYhaCYIWKipiqnyI6fX/g3vg7Oz92t2Z3dk3c+Gw+3Zm7j33/M/33Le5MY5BSmBzkLseN70xAj9QJRiBH4HvXAKPa4Vdok9bWuktzfOZ6EZL8w1qmmVZPKCfE73QIlA25xHN+f2gUGths8sAHoA+Eu1rEXTbOnNfEL08gj+bNnQN/D1i5w/RAdFXBdZw3Vvunp36/rHoWOG553T9bEeKNZs0V3M3Mj4uejosf1f4fEefybDaNfBXA/CHMjKB8cui+0Vvi94P92LNl8L3R/X5a2aOL3Xtd9Gbq5H9ylY1Gf0rDrzXw4jOi34WPRLjrkvgsUQAeagAGsqxV/RhBLgT+u2U6Lbozox4H9S1X0TPi0qeZWUotbywedM7NO+roqZ1I3swQB7IZWJ0CXyNtZtmAuyzoliShjazuTOinNtnvX9im2xZ4H2ZDqAPiq6J7oswhTf4Lvw+ZRBdAW+LxjTR8/i3/tiRYZ57TXP5nuPXlKjkYfoC3CJ8mIdjDkraVCg1o/lW9zzpF+wK+A+0yBuie0WpOttrZNQdBUZP65PYz8gpkgnjpO57bxGprsGzFgJhNbdfC6PcN4FFV8D/FIQXTSwigOaYt1yBx2J5gMcJDb8imoppawDmLCx6QJ/Qg6k+RtJocsCTPBBDnhE9JcIlHxZZ1s11MsdYnPlPv0+5l8bOLEaVtNYDX5oTgVAGxmLeLIJd9r24anIcSrKHRU33DYDInlyHUPZNwAM+a4GfyJFywCPwu0U3RZ+HRS2RsBKMhWGUYbHVXHhNMkY2PwvwpezeLGEeT4ayLjpyISs1NyFqv+g30bsik4nJEwPB8KhqAB4FwOBs1AI/wVutgCxmsxixgoVviaibYRxFMHdjFjoL8LkyjLmvh12WgDfXNk+CtyrgvUJg+Z+EH8hnMLwtEa1uxmMiADRekccDolQe5eebC/jmBDDxWmLBeYDPaS1r+Y3m6nkDfl3rea/khDW6cC+Kms2rWnn4MDkX8E2ry3XS5gG+Nrmrtfh1BR4lJzG28JmqYrx36tTVe4aSbcDgjmqB9+GjFvhScldTRnrX2sfvzbAac+NeOWqBr07umkKxLLxkdVZPl2J8bS3q3VWuWQG/iyR3fVECa0TBTwpU39SqBd5XZNVHr7zwYajkSmvKOd/AyQHqFWSC+QhSAE9czPUPUgD3Ibnz/Xf4TBlPrUxMQaZa4jVZvTFD98zKiJI119bT1rLNhQ/vaUpv6QCv5BX6DDxy+0Fk9XwqtPmWba6plZRvDfAw87WIFyQWWzxDsSNQppG5li0AeLeWKsGs31zq2pkHKXmFvrj0Jh9UJHtEVEu0nGl5MwwjvC49FSubZ3lJM5UkxoBHgAwWsFd9CJMkwycePAtwrwRmfRJiGlkDQu61rCkGjaPdgYcUcNbdytW1fQIdGUGUWezzqMhKN18+E1YBnFNMvrQrvZY1xah6LevjLnHhr8ZiniHA4DVorM5EwADKmHgrFJG8dQEpYbxmpg4ZpMDDrdHKzB366BPw5smQM8OfUmqWz9y7FZTE7yF1WMUMBs8cPefYtHg00PrAPPS6qNk8MAtNXTfGag9icL8dH3opPIxCkaQRYrDk3OkbHmEt2sqlHKBPwJscMaAYqNaMSl23vYAZh1Vo6/4Z5Ib8OIuIh4529Wpi/CLCYmHiVsnqF1mDZ8k9LopK5/MWXWfbPN818ObGOQxpb/XaFh6WQRbctXK1zfdK5+saeDYH+D+KujgCPfQTtnMrzzKAN/C/0Jc2//mhi3/SmFuQ6/bgsoA38O21YhtyIntv69+x2uBnreZYJvBrJZjtzuwI/HZHOLG/EfgR+IFKYKDbHi1+oMD/DyIPdjVz/lGwAAAAAElFTkSuQmCC\" width=\"63\" height=\"18.5\" style=\"width: 63px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t,x] = EulerIntegration(t0,tf,N,x0,func)\r\n  t = 0;\r\n  x = 0;\r\nend","test_suite":"%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) 2*x.*t;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/N;\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    x(i+1) = x(i) + h*func(t(i),x(i));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":4078801,"edited_by":223089,"edited_at":"2025-10-10T13:57:39.000Z","deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":"2025-10-10T13:57:39.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-02-28T23:15:52.000Z","updated_at":"2026-03-23T10:41:43.000Z","published_at":"2024-02-28T23:15:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuler's method approximates the solution to a differential equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + h \\\\cdot f(x, t )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh = \\\\Delta t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's method to integrate over \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equally-spaced points (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equal intervals) between times \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_o\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. You must implement the boundary condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(0)=x0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60983,"title":"Check Euler's characteristic on regular polyhedra","description":"Problem statement\r\nGiven the number of vertices and the number of faces of a given regular polyhedron, compute the number of its edges with Euler characteristic, a powerful generic formula linking these three.\r\n\r\nExamples -\u003e check the test suite\r\n\r\nForbidden functions / expressions\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nMesh processing\r\nMesh generation toolbox","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 414.733px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 207.367px; transform-origin: 408px 207.367px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 263.333px 8px; transform-origin: 263.333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the number of vertices and the number of faces of a given regular polyhedron, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.042px 8px; transform-origin: 110.042px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ecompute the number of its edges\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with Euler characteristic, a powerful generic formula linking these three.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.425px 8px; transform-origin: 6.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.8917px 8px; transform-origin: 59.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echeck the test suite\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/57483\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh processing\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/85173-mesh-generation-toolbox?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh generation toolbox\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function e = check_Euler_chracteristic(v,f)\r\n  e = f;\r\nend","test_suite":"%% tetrahedron\r\nv = 4;\r\nf = 4;\r\ne_correct = 6;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% octahedron\r\nv = 6;\r\nf = 8;\r\ne_correct = 12;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% hexahedron / cube\r\nv = 8;\r\nf = 6;\r\ne_correct = 12;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% icosahedron\r\nv = 12;\r\nf = 20;\r\ne_correct = 30;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% dodecahedron\r\nv = 20;\r\nf = 12;\r\ne_correct = 30;\r\nassert(isequal(check_Euler_chracteristic(v,f),e_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Euler_chracteristic.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:45:16.000Z","deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-23T20:08:07.000Z","updated_at":"2026-02-10T17:11:09.000Z","published_at":"2025-07-24T07:25:21.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the number of vertices and the number of faces of a given regular polyhedron, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ecompute the number of its edges\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with Euler characteristic, a powerful generic formula linking these three.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e-\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003echeck the test suite\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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It's possible to improve on Euler's method by doing the following:\r\n\r\n\r\nThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over  equally-spaced points ( equal intervals) between times  and . You must implement the boundary condition . 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src=\"data:image/png;base64,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\" width=\"44.5\" height=\"18\" style=\"width: 44.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. It's possible to improve on Euler's method by doing the following:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"283.5\" height=\"34.5\" style=\"width: 283.5px; height: 34.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.9167px; text-align: left; transform-origin: 384.5px 31.9167px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equally-spaced points (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39.5\" height=\"18\" style=\"width: 39.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equal intervals) between times \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. You must implement the boundary condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18.5\" style=\"width: 63px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t,x] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n  t = 0;\r\n  x = 0;\r\nend","test_suite":"%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) 2*x*t;\r\nf=func;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/(N-1);\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    xstar = x(i) + h*f(t(i),x(i));\r\n    x(i+1) = x(i) + h/2*(f(t(i),x(i)) + f(t(i+1),xstar));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n\r\n%%\r\nt0=0; tf=1.5; \r\nfunc = @(t,x) x+t;\r\nf=func;\r\nN=50;\r\nt = linspace(t0,tf,N);\r\nh = (tf-t0)/(N-1);\r\nnum_steps = length(t) - 1;\r\nx = zeros(1,num_steps + 1);\r\nx(1) = 1;\r\nfor i = 1:num_steps\r\n    xstar = x(i) + h*f(t(i),x(i));\r\n    x(i+1) = x(i) + h/2*(f(t(i),x(i)) + f(t(i+1),xstar));\r\nend\r\n\r\nx0=1;\r\n[t2,x2] = EulerImprovedIntegration(t0,tf,N,x0,func)\r\n\r\nassert(isequal(x(1),x2(1)))\r\nn1 = length(x);\r\nn2 = length(x2);\r\nassert(isequal(x(n1),x2(n2)))\r\nassert(isequal(n1,n2))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4078801,"edited_by":4078801,"edited_at":"2024-03-05T14:42:54.000Z","deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2024-03-05T14:42:54.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-02-29T19:23:54.000Z","updated_at":"2026-03-31T11:25:40.000Z","published_at":"2024-02-29T19:23:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuler's method approximates the solution to a differential equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + h \\\\cdot f(x, t )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh = \\\\Delta t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It's possible to improve on Euler's method by doing the following:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^* = x(t) + h \\\\cdot f(x,t)\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t+\\\\Delta t) = x(t) + \\\\frac{h}{2} \\\\cdot (f(x,t) + f(t+\\\\Delta t, x^* ) )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe challenge is to write a code that can take some arbitrary function, f(t,x), and use Euler's improved method to integrate over \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equally-spaced points (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equal intervals) between times \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_o\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et_f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. You must implement the boundary condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(0)=x0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60942,"title":"Euler primes","description":"Historical context\r\n\r\n\r\nThe great swiss mathematician Leonhard Euler found the formula  to give some prime numbers.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 41, list the Euler prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 41, your function should return ep = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 41 then ep = 41 and f = 1 since 41 = P(0) and it is the only prime number m greater or equal and less or equal to 41;\r\nIf m = 100 then ep = [41, 43, 47, 53, 61, 71, 83, 97] and f = 8/13 since 41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), and in total there are 13 prime numbers in the range [41; 100] ; \r\nIf m = 19 then ep = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\nn is less or equal to 107 here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\n\r\nSee also\r\nProblem 60944. Ulam primes first formula, Problem 60943. Ulam primes second formula\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 988.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 494.233px; transform-origin: 408px 494.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; 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\"\u003e\u003cspan style=\"\"\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"125\" height=\"19.5\" style=\"width: 125px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euler prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 286.658px 8px; transform-origin: 286.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.9833px 8px; transform-origin: 28.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e41 = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.1167px 8px; transform-origin: 10.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 110.258px 8px; transform-origin: 110.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 8/13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 141.192px 8px; transform-origin: 141.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand in total there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [41; 100] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 19 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3417px 8px; transform-origin: 58.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e107\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60944. Ulam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60943. Ulam primes second formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euler_primes(m)\r\n  ep = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 41;\r\nep_correct = 41;\r\nf_correct = 1;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97];\r\nf_correct = 8/13;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197];\r\nf_correct = 13/34;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 19;\r\nep_correct = [];\r\nf_correct = 0;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euler_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:49:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2025-07-09T05:56:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T14:44:30.000Z","updated_at":"2026-03-16T19:16:06.000Z","published_at":"2025-06-26T15:44:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = n^2 + n + 41}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euler prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 8/13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand in total there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [41; 100] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 19 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e107\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60944. Ulam primes first formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60943. Ulam primes second formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44668,"title":"Day counter function","description":"Write a function called day_counter that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\r\nYou are not allowed to use the datenum built-in function.\r\n(Inspired by Project Euler.)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 144px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 72px; transform-origin: 407px 72px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70px 8px; transform-origin: 70px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function called\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eday_counter\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 265px 8px; transform-origin: 265px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 199px 8px; transform-origin: 199px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYou are not allowed to use the datenum built-in function.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93px 8px; transform-origin: 93px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e(Inspired by Project Euler.)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"% function to count number of mondays\r\nfunction foms = day_counter( year )\r\n\r\nend","test_suite":"%%\r\nyear  = 1902;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2002;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 1996;\r\nfoms_correct = 3;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 1885;\r\nfoms_correct = 1;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2000;\r\nfoms_correct = 1;\r\nassert(isequal(day_counter( year ),foms_correct))\r\n\r\n%%\r\nyear  = 2013;\r\nfoms_correct = 2;\r\nassert(isequal(day_counter( year ),foms_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":171559,"edited_by":223089,"edited_at":"2023-01-16T11:22:13.000Z","deleted_by":null,"deleted_at":null,"solvers_count":77,"test_suite_updated_at":"2023-01-16T11:22:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-05-30T02:42:41.000Z","updated_at":"2026-01-01T20:23:31.000Z","published_at":"2018-05-30T02:42:41.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function called\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eday_counter\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that returns the number of Mondays that fell on the first day of the month in a given year between 1776 and 2016 inclusive where the requested year is the only input to your function and it is a positive integer scalar. Note that a leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. In a leap year, February has 29 days.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYou are not allowed to use the datenum built-in function.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(Inspired by Project Euler.)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42821,"title":"Polygon division","description":"Given the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\r\n\r\n1. The division is done by drawing straight lines between existing vertices.\r\n\r\n2. The triangles are made of existing vertices.\r\n\r\n3. Different orientations of a similar solution are counted as different solutions.\r\n\r\nAssume that n is a positive integer greater than 2.\r\n\r\nExample 1:\r\n\r\nn = 4 (square)\r\n\r\nw = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\r\n\r\nExample 2:\r\n\r\nn = 5 (pentagon)\r\n\r\nw = 5","description_html":"\u003cp\u003eGiven the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\u003c/p\u003e\u003cp\u003e1. The division is done by drawing straight lines between existing vertices.\u003c/p\u003e\u003cp\u003e2. The triangles are made of existing vertices.\u003c/p\u003e\u003cp\u003e3. Different orientations of a similar solution are counted as different solutions.\u003c/p\u003e\u003cp\u003eAssume that n is a positive integer greater than 2.\u003c/p\u003e\u003cp\u003eExample 1:\u003c/p\u003e\u003cp\u003en = 4 (square)\u003c/p\u003e\u003cp\u003ew = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\u003c/p\u003e\u003cp\u003eExample 2:\u003c/p\u003e\u003cp\u003en = 5 (pentagon)\u003c/p\u003e\u003cp\u003ew = 5\u003c/p\u003e","function_template":"function w = polydiv(n)\r\n  w = n;\r\nend","test_suite":"%%\r\nfiletext = fileread('polydiv.m');\r\nassert(isempty(strfind(filetext,'str')))\r\n\r\n%%\r\nn = 3;\r\nw_correct = 1;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 4;\r\nw_correct = 2;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 5;\r\nw_correct = 5;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 8;\r\nw_correct = 132;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 11;\r\nw_correct = 4862;\r\nassert(isequal(polydiv(n),w_correct))\r\n\r\n%%\r\nn = 15;\r\nw_correct = 742900;\r\nassert(isequal(polydiv(n),w_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":15521,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2016-04-30T20:32:42.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2016-04-24T15:28:47.000Z","updated_at":"2026-01-19T17:32:36.000Z","published_at":"2016-04-24T15:28:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the number of vertices (or sides), n, of a planar convex polygon, return the number of ways, w, in which you can divide the polygon into triangles, such that:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e1. The division is done by drawing straight lines between existing vertices.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e2. The triangles are made of existing vertices.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e3. Different orientations of a similar solution are counted as different solutions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that n is a positive integer greater than 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 4 (square)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ew = 2 (you can draw a line between vertices 1 and 3, as well as a line between vertices 2 and 4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample 2:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 5 (pentagon)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ew = 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1483,"title":"Number of paths on a grid","description":"\r\nConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. \r\nYour destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary. \r\n\r\nEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.   \r\n\r\n4x3 has 10 ways\r\n\r\n6x5 has 126 ways\r\n\r\nThis problem can be solved using dynamic programming but there are other methods too. \r\n\r\nProblem 7)\r\nPrev: \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1482 1482\u003e\r\nNext: \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1484 1484\u003e","description_html":"\u003cdiv style = \"text-align: start; line-height: 20px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; transform-origin: 332px 148.5px; vertical-align: baseline; perspective-origin: 332px 148.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; perspective-origin: 309px 63px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. Your destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 21px; white-space: pre-wrap; perspective-origin: 309px 21px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e4x3 has 10 ways\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e6x5 has 126 ways\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis problem can be solved using dynamic programming but there are other methods too.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProblem 7) Prev:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www.mathworks.com/matlabcentral/cody/problems/1482\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e1482\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e Next:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www.mathworks.com/matlabcentral/cody/problems/1484\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e1484\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = paths2dest_ongrid(n,m)\r\n  y = n+m;\r\nend","test_suite":"%%\r\nm = 1; n = 1 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 2; n = 2 ;\r\ny_correct = 2;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 4; n = 3 ;\r\ny_correct = 10;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 6; n = 5 ;\r\ny_correct = 126;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 5; n = 5 ;\r\ny_correct = 70;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 1; n = 100 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 100; n = 1 ;\r\ny_correct = 1;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 2; n = 100 ;\r\ny_correct = 100;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 100; n = 2 ;\r\ny_correct = 100;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n\r\n%%\r\nm = 15; n = 20 ;\r\ny_correct = 818809200;\r\nassert(isequal(paths2dest_ongrid(m,n),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":5,"created_by":11275,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":81,"test_suite_updated_at":"2020-09-28T20:02:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-05-01T14:58:23.000Z","updated_at":"2026-03-19T08:10:31.000Z","published_at":"2013-05-01T14:58:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a grid formed by n vertices vertically down, and m vertices horizontally right. Your starting point is at the top left vertex. Your destination is the bottom right vertex. You are permitted at each vertex to choose to move down or right, that is in the direction towards the destination. You are not to move on what constitutes a back step like moving left or up. If you hit the bottom boundary, or right boundary take it to be given there is only 1 way to the destination, that is following along the boundary.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEx: in a 2x2 grid there are two ways. One way: First down, then right. The other way: First right, then down.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4x3 has 10 ways\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6x5 has 126 ways\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem can be solved using dynamic programming but there are other methods too.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 7) Prev:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1482\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e1482\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e Next:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1484\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e1484\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42938,"title":"Project Euler: Problem 18, Maximum path sum I","description":"By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\r\n\r\n       3*\r\n     7*  4\r\n   2   4*  6\r\n 8   5   9*  3\r\n \r\n3 + 7 + 4 + 9 = 23\r\n\r\nFind the maximum total from top to bottom of a given triangle.\r\n","description_html":"\u003cp\u003eBy starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\u003c/p\u003e\u003cpre\u003e       3*\r\n     7*  4\r\n   2   4*  6\r\n 8   5   9*  3\u003c/pre\u003e\u003cp\u003e3 + 7 + 4 + 9 = 23\u003c/p\u003e\u003cp\u003eFind the maximum total from top to bottom of a given triangle.\u003c/p\u003e","function_template":"function s = maxPathSum(tr)\r\n    % tr: lower triangular matrix\r\n  s = inf;\r\nend","test_suite":"%%\r\nx = [3 0 0 0\r\n  7 4 0 0\r\n  2 4 6 0\r\n  8 5 9 3];\r\nassert(isequal(maxPathSum(x),23))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(magic(10))),891))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(magic(100)^2)),251716502500))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(mod(magic(1000),997))),741570))\r\n\r\n%%\r\nassert(isequal(maxPathSum(tril(gallery('integerdata',1e4,5e3,4))),37455089))","published":true,"deleted":false,"likes_count":4,"comments_count":2,"created_by":85274,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":144,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-08-30T10:52:35.000Z","updated_at":"2026-01-05T00:26:41.000Z","published_at":"2016-08-30T10:53:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBy starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[       3*\\n     7*  4\\n   2   4*  6\\n 8   5   9*  3]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e3 + 7 + 4 + 9 = 23\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the maximum total from top to bottom of a given triangle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45429,"title":"Prime Sum","description":"What is the minimum value that can be written as sum of primes in n different ways?\r\n\r\nFor example,\r\n 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\r\n\r\n \r\n 2 + 2 + 2 + 2 + 2\r\n 2 + 2 + 3 + 3\r\n 2 + 3 + 5\r\n 5 + 5\r\n 3 +7\r\n\r\n\r\n8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\r\n \r\n\r\n\u003chttps://projecteuler.net/problem=77\u003e","description_html":"\u003cp\u003eWhat is the minimum value that can be written as sum of primes in n different ways?\u003c/p\u003e\u003cp\u003eFor example,\r\n 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\u003c/p\u003e\u003cpre\u003e 2 + 2 + 2 + 2 + 2\r\n 2 + 2 + 3 + 3\r\n 2 + 3 + 5\r\n 5 + 5\r\n 3 +7\u003c/pre\u003e\u003cp\u003e8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\u003c/p\u003e\u003cp\u003e\u003ca href = \"https://projecteuler.net/problem=77\"\u003ehttps://projecteuler.net/problem=77\u003c/a\u003e\u003c/p\u003e","function_template":"function i = prime_sum(n)","test_suite":"%%\r\nassert(isequal(prime_sum(3),7))\r\n%%\r\nassert(isequal(prime_sum(6),11))\r\n\r\n%%\r\nassert(isequal(prime_sum(500),45))\r\n\r\n%%\r\nassert(isequal(prime_sum(2678),64))\r\n\r\n%%\r\nassert(isequal(prime_sum(4290),70))\r\n\r\n%%\r\nassert(isequal(prime_sum(6500),75))\r\n\r\n%%\r\nassert(isequal(prime_sum(116500),117))\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":363598,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-04-09T04:25:11.000Z","updated_at":"2020-04-09T04:25:11.000Z","published_at":"2020-04-09T04:25:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the minimum value that can be written as sum of primes in n different ways?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, 10 is the minimum value that can be written as the sum of primes in exactly five different ways:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 2 + 2 + 2 + 2 + 2\\n 2 + 2 + 3 + 3\\n 2 + 3 + 5\\n 5 + 5\\n 3 +7]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 can be written in 3 different ways but 7 can also be written in 3 different ways. So 7 is the right answer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://projecteuler.net/problem=77\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://projecteuler.net/problem=77\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44784,"title":"Project Euler: Problem 14 Longest Collatz sequence(harder version)","description":"This problem is a harder version of \"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\", because of time limits. \u003chttps://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003e \r\n\r\nThe following iterative sequence is defined for the set of positive integers:\r\nn → n/2 (n is even) n → 3n + 1 (n is odd)\r\nUsing the rule above and starting with 13, we generate the following sequence:\r\n13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.\r\nWhich starting number, no more than N, produces the longest chain, and how long? Don't cheat!","description_html":"\u003cp\u003eThis problem is a harder version of \"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\", because of time limits. \u003ca href = \"https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\"\u003ehttps://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003c/a\u003e\u003c/p\u003e\u003cp\u003eThe following iterative sequence is defined for the set of positive integers:\r\nn → n/2 (n is even) n → 3n + 1 (n is odd)\r\nUsing the rule above and starting with 13, we generate the following sequence:\r\n13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.\r\nWhich starting number, no more than N, produces the longest chain, and how long? Don't cheat!\u003c/p\u003e","function_template":"function y = euler014_v2(x)\r\n  y = x;\r\nend","test_suite":"1\r\n%%\r\nassessFunctionAbsence({'tic','toc','pause','etime','clock','now','str2num','timer'},'FileName','euler014_v2.m')\r\n\r\n2\r\n%%\r\nN = 2e8;\r\nnum_correct = 169941673;\r\nlen_correct = 954;\r\ntic\r\n[num, len] = euler014_v2(N);\r\nt=toc\r\nassert(t\u003e1);\r\nassert(isequal(num, num_correct));\r\nassert(isequal(len, len_correct));\r\n\r\n3\r\n%%\r\nN = 4e8;\r\nnum_correct = 268549803;\r\nlen_correct = 965;\r\ntic\r\n[num, len] = euler014_v2(N);\r\nt=toc\r\nassert(t\u003e1);\r\nassert(isequal(num, num_correct));\r\nassert(isequal(len, len_correct));\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":3668,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":"2018-11-11T06:11:05.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-11-09T16:59:43.000Z","updated_at":"2018-11-11T06:11:05.000Z","published_at":"2018-11-09T16:59:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem is a harder version of \\\"Problem 44448. Project Euler: Problem 14 Longest Collatz sequence\\\", because of time limits.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://ww2.mathworks.cn/matlabcentral/cody/problems/44448\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe following iterative sequence is defined for the set of positive integers: n → n/2 (n is even) n → 3n + 1 (n is odd) Using the rule above and starting with 13, we generate the following sequence: 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. 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