{"id":1001,"date":"2020-08-17T23:16:07","date_gmt":"2020-08-17T23:16:07","guid":{"rendered":"http:\/\/blog.atlant.is\/?p=1001"},"modified":"2021-02-21T11:59:00","modified_gmt":"2021-02-21T09:59:00","slug":"schizo-a-new-sudoku-solving-technique","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=1001","title":{"rendered":"Schizo: a different sudoku solving technique"},"content":{"rendered":"\n<p>Over the years, several tips and techniques have been developed to help solve sudoku puzzles with logic rather than brute force. To name a few, <a rel=\"noreferrer noopener\" href=\"https:\/\/sudoku9x9.com\/\" target=\"_blank\">sudoku9x9<\/a> mentions (among others) <em>hidden single<\/em>, <em>naked single<\/em>, <em>x-wing<\/em>. <\/p>\n\n\n\n<!--more-->\n\n\n\n<h2>At least one solution<\/h2>\n\n\n\n<p>These techniques are all based on a common principle: they work \u00ab\u00a0ad absurdum\u00a0\u00bb, by eliminating moves that would lead to an invalid grid: a  puzzle with zero solution. For example, the grid below illustrates the <em>hidden single<\/em> technique:<\/p>\n\n\n\n<div class=\"is-layout-flow wp-block-group\"><div class=\"wp-block-group__inner-container\">\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em;\">\n  <caption><small>Hidden single example<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n  <tbody>\n   <tr> <td style=\"background-color:red;\"> <td style=\"background-color:red;\"> <td style=\"background-color:red;\">   <td>  <td>  <td>2   <td>  <td>  <td> \n   <tr> <td>1 <td style=\"background-color:green;\"> <td>3   <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td style=\"background-color:red;\"> <td style=\"background-color:red;\"> <td style=\"background-color:red;\">   <td>  <td>  <td>    <td>  <td>2 <td> \n  <tbody>\n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n<\/table>\n<\/center>\n\n\n\n<p>Number <strong>2<\/strong> cannot be placed in any of the red cells because this would lead to a grid with no solution. Therefore it must be placed in the green cell, leading to the next step:<\/p>\n\n\n\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em;\">\n  <caption><small>The hidden single is revealed<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n  <tbody>\n   <tr> <td> <td> <td>   <td>  <td>  <td>2   <td>  <td>  <td> \n   <tr> <td>1 <td style=\"color:white;background-color:green\">2 <td>3   <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td> <td> <td>   <td>  <td>  <td>    <td>  <td>2 <td> \n  <tbody>\n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n   <tr> <td>  <td>  <td>    <td>  <td>  <td>    <td>  <td>  <td> \n<\/table>\n<\/center>\n<\/div><\/div>\n\n\n\n<h2>At most one solution<\/h2>\n\n\n\n<p>Here is a (novel?) technique based on the opposite principle: it works by eliminating a move that would lead to a grid with more than one solution. (By current rules, well formed sudoku puzzles must have a single solution.)<\/p>\n\n\n\n<p>[ EDIT &#8212; This technique is in fact well known to people such as the folks at <a href=\"https:\/\/www.youtube.com\/channel\/UCC-UOdK8-mIjxBQm_ot1T-Q\">cracking the cryptic<\/a>. ]<\/p>\n\n\n\n<p>The grid below illustrates this technique, which we call <em>schizo<\/em> (as in schizophrenia):<\/p>\n\n\n\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em;\">\n  <caption><small>Schizo example<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>3 <td>6 <td>1   <td>  <td>2 <td> \n   <tr> <td>2 <td>5 <td>1   <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td style=\"color:white;background-color:green;\"><small>478<\/small>  <td>9   <td>3 <td>  <td>6\n   <tr> <td>6 <td>3 <td>    <td>2 <td style=\"color:white;background-color:red;\"><small>478<\/small>  <td>5   <td>  <td>1 <td>9\n  <tbody>\n   <tr> <td>  <td>2 <td>    <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td>7   <td>1 <td>  <td> \n   <tr> <td>7 <td>1 <td>9   <td>5 <td>3 <td>6   <td>2 <td>8 <td>4\n   <tr> <td>  <td>  <td>    <td>1 <td>9 <td>2   <td>  <td>  <td> \n  <tbody>\n   <tr> <td>  <td>4 <td>2   <td>7 <td>1 <td>3   <td>  <td>5 <td> \n   <tr> <td>1 <td>  <td>    <td>9 <td>5 <td>8   <td>  <td>  <td>2\n   <tr> <td>5 <td>  <td>    <td>6 <td>2 <td>4   <td>  <td>  <td>1\n<\/table>\n<\/center>\n\n\n\n<p>Number <strong>7<\/strong> cannot be placed in the red cell because this would eventually lead to two equally valid solutions:<\/p>\n\n\n\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em\">\n  <caption><small>Eventual solution 1<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>3 <td>6 <td>1   <td>  <td>2 <td> \n   <tr> <td>2 <td>5 <td>1   <td style=\"color:white;background-color:blue;\">4  <td style=\"color:white;background-color:blue;\">8  <td>9   <td>3 <td>  <td>6\n   <tr> <td>6 <td>3 <td>    <td>2 <td style=\"color:white;background-color:red;\">7  <td>5   <td>  <td>1 <td>9\n  <tbody>\n   <tr> <td>  <td>2 <td>    <td style=\"color:white;background-color:blue;\">8  <td style=\"color:white;background-color:blue;\">4  <td>7   <td>1 <td>  <td> \n   <tr> <td>7 <td>1 <td>9   <td>5 <td>3 <td>6   <td>2 <td>8 <td>4\n   <tr> <td>  <td>  <td>    <td>1 <td>9 <td>2   <td>  <td>  <td> \n  <tbody>\n   <tr> <td>  <td>4 <td>2   <td>7 <td>1 <td>3   <td>  <td>5 <td> \n   <tr> <td>1 <td>  <td>    <td>9 <td>5 <td>8   <td>  <td>  <td>2\n   <tr> <td>5 <td>  <td>    <td>6 <td>2 <td>4   <td>  <td>  <td>1\n<\/table>\n<\/center>\n\n\n\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em;float:center\">\n  <caption><small>Eventual solution 2<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>3 <td>6 <td>1   <td>  <td>2 <td> \n   <tr> <td>2 <td>5 <td>1   <td style=\"color:white;background-color:blue;\">8  <td style=\"color:white;background-color:blue;\">4  <td>9   <td>3 <td>  <td>6\n   <tr> <td>6 <td>3 <td>    <td>2 <td style=\"color:white;background-color:red;\">7  <td>5   <td>  <td>1 <td>9\n  <tbody>\n   <tr> <td>  <td>2 <td>    <td style=\"color:white;background-color:blue;\">4  <td style=\"color:white;background-color:blue;\">8  <td>7   <td>1 <td>  <td> \n   <tr> <td>7 <td>1 <td>9   <td>5 <td>3 <td>6   <td>2 <td>8 <td>4\n   <tr> <td>  <td>  <td>    <td>1 <td>9 <td>2   <td>  <td>  <td> \n  <tbody>\n   <tr> <td>  <td>4 <td>2   <td>7 <td>1 <td>3   <td>  <td>5 <td> \n   <tr> <td>1 <td>  <td>    <td>9 <td>5 <td>8   <td>  <td>  <td>2\n   <tr> <td>5 <td>  <td>    <td>6 <td>2 <td>4   <td>  <td>  <td>1\n<\/table>\n<\/center>\n\n\n\n<p>Therefore <strong>7<\/strong> must be placed in the green cell, leading to the next step:<\/p>\n\n\n\n<style>\ntable { border-collapse: collapse; font-family: Calibri, sans-serif; }\ncolgroup, tbody { border: solid medium; }\ntd { border: solid thin; height: 2em; width: 2em; text-align: center; padding: 0; }\n<\/style>\n<center>\n<table style=\"width:18em;\">\n  <caption><small>Schizophrenia is avoided<\/small><\/caption>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <colgroup><col><col><col><\/colgroup>\n  <tbody>\n   <tr> <td>  <td>  <td>    <td>3 <td>6 <td>1   <td>  <td>2 <td> \n   <tr> <td>2 <td>5 <td>1   <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td style=\"color:white;background-color:green;\">7  <td>9   <td>3 <td>  <td>6\n   <tr> <td>6 <td>3 <td>    <td>2 <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td>5   <td>  <td>1 <td>9\n  <tbody>\n   <tr> <td>  <td>2 <td>    <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td style=\"color:white;background-color:blue;\"><small>48<\/small>  <td>7   <td>1 <td>  <td> \n   <tr> <td>7 <td>1 <td>9   <td>5 <td>3 <td>6   <td>2 <td>8 <td>4\n   <tr> <td>  <td>  <td>    <td>1 <td>9 <td>2   <td>  <td>  <td> \n  <tbody>\n   <tr> <td>  <td>4 <td>2   <td>7 <td>1 <td>3   <td>  <td>5 <td> \n   <tr> <td>1 <td>  <td>    <td>9 <td>5 <td>8   <td>  <td>  <td>2\n   <tr> <td>5 <td>  <td>    <td>6 <td>2 <td>4   <td>  <td>  <td>1\n<\/table>\n<\/center>\n\n\n\n<p>$\\Box$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Over the years, several tips and techniques have been developed to help solve sudoku puzzles with logic rather than brute force. To name a few, sudoku9x9 mentions (among others) hidden single, naked single, x-wing.<\/p>\n","protected":false},"author":1,"featured_media":3439,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[16,19,21],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/1001"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1001"}],"version-history":[{"count":52,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/1001\/revisions"}],"predecessor-version":[{"id":3440,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/1001\/revisions\/3440"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/3439"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1001"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1001"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1001"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=1001"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}