{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","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":"2026-04-16T00:00:00.000Z","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":2340,"title":"Numbers spiral diagonals (Part 1)","description":"Inspired by Project Euler n°28 et 58.\r\n\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\n\r\nFor exemple with n=5, the spiral matrix is :\r\n\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\n\r\nIn this example, the sum of the numbers on the diagonals is 101.\r\n\r\nWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\r\n\r\nHINTS: You want the diagonals, not the whole matrix.","description_html":"\u003cp\u003eInspired by Project Euler n°28 et 58.\u003c/p\u003e\u003cp\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/p\u003e\u003cp\u003eFor exemple with n=5, the spiral matrix is :\u003c/p\u003e\u003cpre\u003e                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\u003c/pre\u003e\u003cp\u003eIn this example, the sum of the numbers on the diagonals is 101.\u003c/p\u003e\u003cp\u003eWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\u003c/p\u003e\u003cp\u003eHINTS: You want the diagonals, not the whole matrix.\u003c/p\u003e","function_template":"function y = spiral_nb(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 3;\r\ny_correct = 25;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 5;\r\ny_correct = 101;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 9;\r\ny_correct = 537;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 501;\r\ny_correct = 83960501;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 5001;\r\ny_correct = 83395855001;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 10001;\r\ny_correct = 666916710001;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 10003;\r\ny_correct = 667316890025;\r\nassert(isequal(spiral_nb(x),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":5390,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":297,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-30T22:02:51.000Z","updated_at":"2026-04-21T16:06:00.000Z","published_at":"2014-05-30T22:03:01.000Z","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\u003eInspired by Project Euler n°28 et 58.\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\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\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 exemple with n=5, the spiral matrix is :\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[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\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\u003eIn this example, the sum of the numbers on the diagonals is 101.\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\u003eWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\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\u003eHINTS: You want the diagonals, not the whole matrix.\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":2342,"title":"Numbers spiral diagonals (Part 2)","description":"Inspired by Project Euler n°28 and 58.\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\nFor example with n=5, the spiral matrix is :\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\nThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\r\nWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\r\nWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u003cp\u003c1)","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: 326.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 163.083px; transform-origin: 407px 163.083px; 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: 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: 120px 8px; transform-origin: 120px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°28 and 58.\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; 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border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       19  6  1  2 11\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); 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border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       18  5  4  3 12\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       17 16 15 14 13\u003c/span\u003e\u003c/span\u003e\u003c/div\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: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; 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: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\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: 262px 8px; transform-origin: 262px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\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: 384px 21px; text-align: left; transform-origin: 384px 21px; 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: 364.5px 8px; transform-origin: 364.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"perspective-origin: 74px 8px; transform-origin: 74px 8px; \"\u003eWhat is the side length \u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e(\u003c/span\u003e\u003cspan style=\"perspective-origin: 96px 8px; transform-origin: 96px 8px; \"\u003ealways odd and greater than 1\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e)\u003c/span\u003e\u003cspan style=\"perspective-origin: 189.5px 8px; transform-origin: 189.5px 8px; \"\u003e of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function res=spiral_ratio(pourcentage)\r\nres=pourcentage*2;\r\nend","test_suite":"%%\r\nx = 0.8;\r\ny_correct = 3;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.5;\r\ny_correct = 11;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.4;\r\ny_correct = 31;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.3;\r\ny_correct = 49;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.25;\r\ny_correct = 99;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.2;\r\ny_correct = 309;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.15;\r\ny_correct = 981;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.14;\r\ny_correct = 1883;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.13;\r\ny_correct = 3593;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.12;\r\ny_correct = 6523;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.11;\r\ny_correct = 12201;\r\nassert(isequal(spiral_ratio(x),y_correct))","published":true,"deleted":false,"likes_count":7,"comments_count":5,"created_by":5390,"edited_by":223089,"edited_at":"2022-09-26T17:42:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":199,"test_suite_updated_at":"2022-07-09T19:28:50.000Z","rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-31T18:36:25.000Z","updated_at":"2026-04-22T04:16:05.000Z","published_at":"2014-05-31T18:53:35.000Z","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\u003eInspired by Project Euler n°28 and 58.\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\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\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 example with n=5, the spiral matrix is :\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[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\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 sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\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\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal 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\u003eWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\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\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":2340,"title":"Numbers spiral diagonals (Part 1)","description":"Inspired by Project Euler n°28 et 58.\r\n\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\n\r\nFor exemple with n=5, the spiral matrix is :\r\n\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\n\r\nIn this example, the sum of the numbers on the diagonals is 101.\r\n\r\nWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\r\n\r\nHINTS: You want the diagonals, not the whole matrix.","description_html":"\u003cp\u003eInspired by Project Euler n°28 et 58.\u003c/p\u003e\u003cp\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/p\u003e\u003cp\u003eFor exemple with n=5, the spiral matrix is :\u003c/p\u003e\u003cpre\u003e                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\u003c/pre\u003e\u003cp\u003eIn this example, the sum of the numbers on the diagonals is 101.\u003c/p\u003e\u003cp\u003eWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\u003c/p\u003e\u003cp\u003eHINTS: You want the diagonals, not the whole matrix.\u003c/p\u003e","function_template":"function y = spiral_nb(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 3;\r\ny_correct = 25;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 5;\r\ny_correct = 101;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 9;\r\ny_correct = 537;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 501;\r\ny_correct = 83960501;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 5001;\r\ny_correct = 83395855001;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 10001;\r\ny_correct = 666916710001;\r\nassert(isequal(spiral_nb(x),y_correct))\r\n%%\r\nx = 10003;\r\ny_correct = 667316890025;\r\nassert(isequal(spiral_nb(x),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":5390,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":297,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-30T22:02:51.000Z","updated_at":"2026-04-21T16:06:00.000Z","published_at":"2014-05-30T22:03:01.000Z","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\u003eInspired by Project Euler n°28 et 58.\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\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\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 exemple with n=5, the spiral matrix is :\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[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\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\u003eIn this example, the sum of the numbers on the diagonals is 101.\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\u003eWhat is the sum of the numbers on the diagonals in any n by n spiral (n always odd) ?\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\u003eHINTS: You want the diagonals, not the whole matrix.\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":2342,"title":"Numbers spiral diagonals (Part 2)","description":"Inspired by Project Euler n°28 and 58.\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\nFor example with n=5, the spiral matrix is :\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\nThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\r\nWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\r\nWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u003cp\u003c1)","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: 326.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 163.083px; transform-origin: 407px 163.083px; 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: 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: 120px 8px; transform-origin: 120px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°28 and 58.\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: 341px 8px; transform-origin: 341px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\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: 131.5px 8px; transform-origin: 131.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example with n=5, the spiral matrix is :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 51.0833px; transform-origin: 404px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       21 22 23 24 25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       20  7  8  9 10\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       19  6  1  2 11\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       18  5  4  3 12\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       17 16 15 14 13\u003c/span\u003e\u003c/span\u003e\u003c/div\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: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; 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: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\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: 262px 8px; transform-origin: 262px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\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: 384px 21px; text-align: left; transform-origin: 384px 21px; 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: 364.5px 8px; transform-origin: 364.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"perspective-origin: 74px 8px; transform-origin: 74px 8px; \"\u003eWhat is the side length \u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e(\u003c/span\u003e\u003cspan style=\"perspective-origin: 96px 8px; transform-origin: 96px 8px; \"\u003ealways odd and greater than 1\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e)\u003c/span\u003e\u003cspan style=\"perspective-origin: 189.5px 8px; transform-origin: 189.5px 8px; \"\u003e of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function res=spiral_ratio(pourcentage)\r\nres=pourcentage*2;\r\nend","test_suite":"%%\r\nx = 0.8;\r\ny_correct = 3;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.5;\r\ny_correct = 11;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.4;\r\ny_correct = 31;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.3;\r\ny_correct = 49;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.25;\r\ny_correct = 99;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.2;\r\ny_correct = 309;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.15;\r\ny_correct = 981;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.14;\r\ny_correct = 1883;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.13;\r\ny_correct = 3593;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.12;\r\ny_correct = 6523;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.11;\r\ny_correct = 12201;\r\nassert(isequal(spiral_ratio(x),y_correct))","published":true,"deleted":false,"likes_count":7,"comments_count":5,"created_by":5390,"edited_by":223089,"edited_at":"2022-09-26T17:42:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":199,"test_suite_updated_at":"2022-07-09T19:28:50.000Z","rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-31T18:36:25.000Z","updated_at":"2026-04-22T04:16:05.000Z","published_at":"2014-05-31T18:53:35.000Z","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\u003eInspired by Project Euler n°28 and 58.\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\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\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 example with n=5, the spiral matrix is :\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[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\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 sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\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\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal 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\u003eWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? 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