{"id":1097,"date":"2026-06-29T10:16:11","date_gmt":"2026-06-28T22:16:11","guid":{"rendered":"https:\/\/thesudoku.com\/blog\/?page_id=1097"},"modified":"2026-08-27T14:12:46","modified_gmt":"2026-08-27T02:12:46","slug":"sudoku-rules-loops-colours","status":"publish","type":"page","link":"https:\/\/thesudoku.com\/blog\/sudoku-rules-loops-colours\/","title":{"rendered":"Sudoku Rules Loops &amp; Colours"},"content":{"rendered":"\n<div class=\"wp-block-group content-wrapper wp-container-content-69bc4bdf has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-left-radius:0px;border-bottom-right-radius:0px;margin-top:var(--wp--preset--spacing--50);margin-bottom:var(--wp--preset--spacing--50)\">\n<div class=\"wp-block-group has-base-background-color has-background has-global-padding is-layout-constrained wp-container-core-group-is-layout-3b2d7e69 wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-left-radius:0px;border-bottom-right-radius:0px;min-height:70vh;margin-top:0;margin-bottom:0;padding-top:var(--wp--preset--spacing--50);padding-right:var(--wp--preset--spacing--30);padding-bottom:var(--wp--preset--spacing--50);padding-left:var(--wp--preset--spacing--30)\">\n<h1 class=\"wp-block-heading is-style-text-display is-style-text-display--1\" style=\"font-size:clamp(17.905px, 1.119rem + ((1vw - 3.2px) * 0.99), 28px);font-style:normal;font-weight:1000\">Sudoku Rules<\/h1>\n\n\n\n<div class=\"wp-block-group is-layout-flex wp-block-group-is-layout-flex\" style=\"padding-top:var(--wp--preset--spacing--20);padding-bottom:var(--wp--preset--spacing--20)\">\n<p style=\"border-style:none;border-width:0px;font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:400\"><a href=\"https:\/\/thesudoku.com\/sudoku-rules\/\" data-type=\"page\" data-id=\"73\">Easy<\/a><\/p>\n\n\n\n<p style=\"border-style:none;border-width:0px;font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);\"><a href=\"https:\/\/thesudoku.com\/sudoku-rules-normal\/\" data-type=\"page\" data-id=\"73\">Normal<\/a><\/p>\n\n\n\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);\"><a href=\"https:\/\/thesudoku.com\/sudoku-rules-hard\/\" data-type=\"page\" data-id=\"136\">Hard<\/a><\/p>\n\n\n\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);\"><a href=\"https:\/\/thesudoku.com\/sudoku-rules-expert\/\" data-type=\"page\" data-id=\"139\">Expert<\/a><\/p>\n\n\n\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);\"><a href=\"https:\/\/thesudoku.com\/sudoku-rules-master\/\" data-type=\"page\" data-id=\"142\">Master<\/a><\/p>\n\n\n\n<div class=\"wp-block-group has-custom-d-9-e-7-f-9-background-color has-background has-global-padding is-layout-constrained wp-container-core-group-is-layout-25dcf7c9 wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:100px;border-top-right-radius:100px;border-bottom-left-radius:100px;border-bottom-right-radius:100px;padding-top:5px;padding-right:var(--wp--preset--spacing--30);padding-bottom:5px;padding-left:var(--wp--preset--spacing--30)\">\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:700\"><a data-type=\"page\" data-id=\"139\" href=\"https:\/\/thesudoku.com\/sudoku-rules-expert\/\">Chains &amp; Colours<\/a><\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-style-default is-layout-grid wp-container-core-group-is-layout-a10ea86b wp-block-group-is-layout-grid\" style=\"min-height:300px;margin-top:0;margin-bottom:0;padding-top:0;padding-right:0px;padding-bottom:0;padding-left:0px\">\n<div class=\"wp-block-group hover-anim has-custom-fcfcfc-background-color has-background tw-stretched-link has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:4px;border-top-right-radius:4px;border-bottom-left-radius:4px;border-bottom-right-radius:4px;min-height:300px\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\" style=\"margin-top:40px;margin-right:0;margin-bottom:40px;margin-left:0\"><a href=\"https:\/\/thesudoku.com\/blog\/3d-medusa\/\"><img loading=\"lazy\" decoding=\"async\" width=\"248\" height=\"240\" src=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6505.png\" alt=\"\" class=\"wp-image-1099\" style=\"width:120px\"\/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-custom-e-7-e-8-ea-color has-alpha-channel-opacity has-custom-e-7-e-8-ea-background-color has-background my-margin\"\/>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-f0a7149d wp-block-group-is-layout-constrained\" style=\"margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:0;padding-right:var(--wp--preset--spacing--30);padding-bottom:0;padding-left:var(--wp--preset--spacing--30)\">\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:600\">3D Medusa<\/p>\n\n\n\n<p class=\"my-margin\" style=\"font-size:14px\">Advanced multi-digit coloring across interconnected candidates<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group hover-anim has-custom-fcfcfc-background-color has-background tw-stretched-link wp-container-content-da6b9471 has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-style:none;border-width:0px;border-top-left-radius:4px;border-top-right-radius:4px;border-bottom-left-radius:4px;border-bottom-right-radius:4px;min-height:300px;margin-top:0;margin-bottom:0\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\" style=\"margin-top:40px;margin-right:0;margin-bottom:40px;margin-left:0\"><a href=\"https:\/\/thesudoku.com\/blog\/dual-medusa\/\"><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"240\" src=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30222.png\" alt=\"\" class=\"wp-image-1098\" style=\"width:120px\" srcset=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30222.png 240w, https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30222-150x150.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-custom-e-7-e-8-ea-color has-alpha-channel-opacity has-custom-e-7-e-8-ea-background-color has-background my-margin\"\/>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-f0a7149d wp-block-group-is-layout-constrained\" style=\"margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:0;padding-right:var(--wp--preset--spacing--30);padding-bottom:0;padding-left:var(--wp--preset--spacing--30)\">\n<p class=\"has-text-align-left\" style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:600\"><a href=\"https:\/\/thesudoku.com\/blog\/dual-medusa\/\">Dual Medusa<\/a><\/p>\n\n\n\n<p class=\"my-margin\" style=\"font-size:14px\">Two-coloring using a single digit<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group hover-anim has-custom-fcfcfc-background-color has-background tw-stretched-link has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:4px;border-top-right-radius:4px;border-bottom-left-radius:4px;border-bottom-right-radius:4px;min-height:300px\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\" style=\"margin-top:40px;margin-right:0;margin-bottom:40px;margin-left:0\"><a href=\"https:\/\/thesudoku.com\/blog\/forcing-chains\/\"><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"240\" src=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6503.png\" alt=\"\" class=\"wp-image-1100\" style=\"width:120px\" srcset=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6503.png 240w, https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6503-150x150.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-custom-e-7-e-8-ea-color has-alpha-channel-opacity has-custom-e-7-e-8-ea-background-color has-background my-margin\"\/>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-f0a7149d wp-block-group-is-layout-constrained\" style=\"margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:0;padding-right:var(--wp--preset--spacing--30);padding-bottom:0;padding-left:var(--wp--preset--spacing--30)\">\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:600\">Forcing Chains<\/p>\n\n\n\n<p class=\"my-margin\" style=\"font-size:14px\">Following logical consequences from an assumed candidate<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group hover-anim has-custom-fcfcfc-background-color has-background tw-stretched-link has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:4px;border-top-right-radius:4px;border-bottom-left-radius:4px;border-bottom-right-radius:4px;min-height:300px\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\" style=\"margin-top:40px;margin-right:0;margin-bottom:40px;margin-left:0\"><a href=\"https:\/\/thesudoku.com\/blog\/nishio\/\"><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"240\" src=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30221@2x.png\" alt=\"\" class=\"wp-image-1101\" style=\"width:120px\" srcset=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30221@2x.png 240w, https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-30221@2x-150x150.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-custom-e-7-e-8-ea-color has-alpha-channel-opacity has-custom-e-7-e-8-ea-background-color has-background my-margin\"\/>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-f0a7149d wp-block-group-is-layout-constrained\" style=\"margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:0;padding-right:var(--wp--preset--spacing--30);padding-bottom:0;padding-left:var(--wp--preset--spacing--30)\">\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:600\">Nishio<\/p>\n\n\n\n<p class=\"my-margin\" style=\"font-size:14px\">Trial a candidate and eliminate it if it leads to a contradiction<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group hover-anim has-custom-fcfcfc-background-color has-background tw-stretched-link has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-top-left-radius:4px;border-top-right-radius:4px;border-bottom-left-radius:4px;border-bottom-right-radius:4px;min-height:300px\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\" style=\"margin-top:40px;margin-right:0;margin-bottom:40px;margin-left:0\"><a href=\"https:\/\/thesudoku.com\/blog\/dynamic-forcing-chains\/\"><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"240\" src=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6504.png\" alt=\"\" class=\"wp-image-1102\" style=\"width:120px\" srcset=\"https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6504.png 240w, https:\/\/thesudoku.com\/blog\/wp-content\/uploads\/2026\/06\/Group-6504-150x150.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-custom-e-7-e-8-ea-color has-alpha-channel-opacity has-custom-e-7-e-8-ea-background-color has-background my-margin\"\/>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-f0a7149d wp-block-group-is-layout-constrained\" style=\"margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:0;padding-right:var(--wp--preset--spacing--30);padding-bottom:0;padding-left:var(--wp--preset--spacing--30)\">\n<p style=\"font-size:clamp(14px, 0.875rem + ((1vw - 3.2px) * 0.196), 16px);font-style:normal;font-weight:600\">Dynamic Forcing Chains<\/p>\n\n\n\n<p class=\"my-margin\" style=\"font-size:14px\">Forcing chains that branch through multiple logical possibilities<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-container-core-group-is-layout-ff778368 wp-block-group-is-layout-constrained\" style=\"padding-right:var(--wp--preset--spacing--30);padding-left:var(--wp--preset--spacing--30)\">\n<h2 class=\"wp-block-heading\">5 Ultimate Chaining &amp; Coloring Techniques<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><a href=\"https:\/\/thesudoku.com\/3d-medusa\/\">1. 3D Medusa (Multi-Digit Coloring)<\/a><\/strong><\/h3>\n\n\n\n<p><strong>3D Medusa<\/strong> extends basic 2-color Simple Coloring by tracking candidates across <strong>both bi-location links<\/strong> (a candidate appearing twice in a row, column, or block) and <strong>bi-value cells<\/strong> (a cell containing exactly two candidates).<\/p>\n\n\n\n<p>By alternating two colors (e.g., <strong>Green<\/strong> = True, <strong>Blue<\/strong> = False) through strong links across different digits, you build a multi-digit web. Elimination occurs when logic breaks:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Twice in a Unit:<\/strong> If two candidates of the <em>same color<\/em> share a unit, that entire color set is false.<\/li>\n\n\n\n<li><strong>Two Colors in a Cell:<\/strong> If a single cell contains both colors, the blue\/green assignment for that cell forces an instant collapse.<\/li>\n\n\n\n<li><strong>Emptied Cell:<\/strong> If all candidates in a cell share the same color, that color must be false (otherwise the cell is left empty).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><a href=\"https:\/\/thesudoku.com\/dual-medusa\/\">2. Dual Medusa<\/a><\/strong><\/h3>\n\n\n\n<p><strong>Dual Medusa<\/strong> applies when a grid contains two completely separate 3D Medusa coloring webs that do not directly intersect through strong links. Instead of analyzing them in isolation, you look at how the two independent webs interact within shared units. If a candidate outside both chains sees a specific color state from Web A <em>and<\/em> a specific color state from Web B\u2014and both states force the elimination of that candidate\u2014it can be safely erased.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><a href=\"https:\/\/thesudoku.com\/nishio\/\">3. Nishio (Nishio Forcing Chains)<\/a><\/strong><\/h3>\n\n\n\n<p><strong>Nishio<\/strong> is a targeted proof-by-contradiction technique focused on a <strong>single candidate placement<\/strong>. You select a cell with a candidate, assume it is <strong>TRUE<\/strong>, and follow the immediate chain reaction of strong and weak links. If this initial placement forces an illegal grid state (such as leaving a row, column, or block without any valid placement for that digit), your initial assumption was false. You can immediately erase that candidate from the starting cell.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><a href=\"https:\/\/thesudoku.com\/forcing-chains\/\">4. Forcing Chains<\/a><\/strong><\/h3>\n\n\n\n<p><strong>Forcing Chains<\/strong> explore all possible choices for a specific starting condition (such as every candidate in a tri-value cell, or every location for a digit in a row) to see if they all lead to the exact same outcome.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Cell Forcing Chains:<\/strong> If a cell has candidates <code>[1, 4, 7]<\/code>, you build three separate chains assuming <code>Cell=1<\/code>, <code>Cell=4<\/code>, and <code>Cell=7<\/code>. If all three independent paths force cell <code>X9<\/code> to be <code>5<\/code>, then <code>X9=5<\/code> is mathematically guaranteed.<\/li>\n\n\n\n<li><strong>Region Forcing Chains:<\/strong> If candidate <code>3<\/code> can only go in three spots in Row 1, you trace chains from all three spots. Any candidate eliminated by <em>all<\/em> branch paths can be permanently deleted.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong><a href=\"https:\/\/thesudoku.com\/dynamic-forcing-chains\/\">5. Dynamic Forcing Chains<\/a><\/strong><\/h3>\n\n\n\n<p>The absolute ceiling of Sudoku logic. <strong>Dynamic Forcing Chains<\/strong> allow chains to split, nest inside other chains, and dynamically update the rules of the grid as hypothetical placements are made. Unlike static forcing chains\u2014which follow simple alternating logic\u2014dynamic chains can discover complex internal sub-contradictions within individual branches. If a sub-branch creates a mini-loop or triggers a sub-forcing chain, the master chain incorporates that logical deduction to clear candidates across the grid.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Master the Ultimate Chain Logic<\/h2>\n\n\n\n<p>Mastering loops, coloring webs, and forcing chains transforms extreme-level Sudoku from overwhelming candidate clutter into structured logical paths. Systematically label your strong links, alternate your colors, and test branch implications to dismantle the toughest puzzles available.<\/p>\n\n\n\n<p><strong>Ready to test your chaining skills?<\/strong> Dive into our <strong>Master Sudoku Puzzles<\/strong> now to apply 3D Medusa, Nishio, and Forcing Chains in real time!<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Sudoku Rules Easy Normal Hard Expert Master Chains &amp; Colours 3D Medusa Advanced multi-digit coloring across interconnected candidates Dual Medusa Two-coloring using a single digit Forcing Chains Following logical consequences from an assumed candidate Nishio Trial a candidate and eliminate it if it leads to a contradiction Dynamic Forcing Chains Forcing chains that branch through [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-no-title","meta":{"footnotes":""},"categories":[116],"tags":[],"class_list":["post-1097","page","type-page","status-publish","hentry","category-master-techniques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Sudoku Rules Loops &amp; 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