{"id":35560,"date":"2026-04-20T00:00:48","date_gmt":"2026-04-20T07:00:48","guid":{"rendered":"https:\/\/www.gmpuzzles.com\/blog\/?p=35560"},"modified":"2026-04-18T10:11:01","modified_gmt":"2026-04-18T17:11:01","slug":"kropki-cages-sudoku-by-bill-murphy","status":"publish","type":"post","link":"https:\/\/www.gmpuzzles.com\/blog\/2026\/04\/kropki-cages-sudoku-by-bill-murphy\/","title":{"rendered":"Kropki Cages Sudoku by Bill Murphy"},"content":{"rendered":"<p>(This post is part of the <a href=\"https:\/\/www.gmpuzzles.com\/blog\/tag\/gas\/\">Genuinely Approachable Sudoku (GAS)<\/a> series.)<br \/>\n<em>In this twist on the typical Kropki formula, it is the sums of the values in each cage that are being compared when considering the black and white dots.<\/em><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/www.gmpuzzles.com\/images\/puzzles2026\/260420-KropkiCagesSudoku.png\" alt=\"Kropki Cages Sudoku by Bill Murphy\" \/><\/p>\n<p align=\"center\"><a href=\"https:\/\/www.gmpuzzles.com\/images\/puzzles2026\/260420-KropkiCagesSudoku.pdf\">PDF<\/a><\/p>\n<p>or <a href=\"https:\/\/gmpuzzles.com\/s\/260420Krop\">solve online<\/a> (using SudokuPad)<\/p>\n<p><b>Author\/Opus:<\/b> This is the 107th puzzle from Bill Murphy, part of the Genuinely Approachable Sudoku (<a href=\"https:\/\/sudokutheory.com\/gas\/\" \/>GAS<\/a>) team.<\/p>\n<p><b>Rules:<\/b> Standard Sudoku rules (insert a number from 1 to 9 into each cell so that no number repeats in any row, column, or bold region). If a white circle is given between two cages, then the sums of the numbers in those cages must differ by 1. If a black circle is given between two cages, then the sums of the numbers in those cages must have a ratio of 2. Numbers cannot repeat in a cage, and not all circles are given.<\/p>\n<p><b>GAS Time Standards (highlight to view):<\/b> Two party hats (\ud83c\udfa9\ud83e\udd73): <span style=\"color: white;background-color: white\">8:00<\/span>; One party hat (\ud83e\udd73): <span style=\"color: white;background-color: white\">15:00<\/span>. All other solvers earn a \ud83e\udd95: <span style=\"color: white;background-color: white\">C-Saurophaganax<\/span>.<\/p>\n<p><b>Thomas Hits the GAS (highlight to view): <\/b> <span style=\"color: white;background-color: white\">2:50<\/span>, with <a href=\"https:\/\/www.gmpuzzles.com\/images\/puzzles2026\/sudokupad-Bill_Murph-Kropki_Cages_Sudoku-a633-d171.replay\">SudokuPad replay file<\/a> shared as a download for now (requires loading via settings menu with improvements expected before people should use this regularly).<\/p>\n<p><b>Solution:<\/b> <a href=\"https:\/\/www.gmpuzzles.com\/images\/puzzles2026\/260420-KropkiCagesSudoku-soln.pdf\">PDF<\/a>; solving video with explanation to be added later from GAS team.<\/p>\n<p><b>Note:<\/b> Follow this link for other less common variations of <a href=\"https:\/\/www.gmpuzzles.com\/blog\/tag\/sudoku-3+variation-2\/\">Sudoku<\/a>.<\/p>\n<p><b>Note 2:<\/b>\u00a0Comments on the blog are great! For a more interactive discussion, please also consider using our\u00a0<a href=\"https:\/\/discord.com\/channels\/1059513302847336469\/1380897111599616020\">Daily GAS discussion<\/a>\u00a0post\u00a0on the GMPuzzles Discord. Not a member of the Discord? Click\u00a0<a href=\"https:\/\/discord.gg\/RNNfEBEbQS\">this link<\/a>\u00a0for basic access.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(This post is part of the Genuinely Approachable Sudoku (GAS) series.) In this twist on the typical Kropki formula, it is the sums of the values in each cage that are being compared when considering the black and white dots. PDF or solve online (using SudokuPad) Author\/Opus: This is the 107th puzzle from Bill Murphy, [&hellip;]<\/p>\n","protected":false},"author":443,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,126],"tags":[8,15,56,237],"class_list":["post-35560","post","type-post","status-publish","format-standard","hentry","category-puzzle","category-othersudoku","tag-sudoku-3","tag-wednesdaythursday","tag-variation-2","tag-gas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Kropki Cages Sudoku by Bill Murphy - The Art of Puzzles<\/title>\n<meta name=\"description\" content=\"In this twist on the Kropki formula, it is the sums of the values in each cage that are being compared when considering the colored dots.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.gmpuzzles.com\/blog\/2026\/04\/kropki-cages-sudoku-by-bill-murphy\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Kropki Cages Sudoku by Bill Murphy - The Art of Puzzles\" \/>\n<meta property=\"og:description\" content=\"In this twist on the Kropki formula, it is the sums of the values in each cage that are being compared when considering the colored dots.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.gmpuzzles.com\/blog\/2026\/04\/kropki-cages-sudoku-by-bill-murphy\/\" \/>\n<meta property=\"og:site_name\" content=\"The Art of Puzzles\" \/>\n<meta property=\"article:published_time\" content=\"2026-04-20T07:00:48+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.gmpuzzles.com\/images\/puzzles2026\/260420-KropkiCagesSudoku.png\" \/>\n<meta name=\"author\" content=\"Bill Murphy\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@drsudoku\" \/>\n<meta name=\"twitter:site\" content=\"@drsudoku\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Bill Murphy\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/\"},\"author\":{\"name\":\"Bill Murphy\",\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/#\\\/schema\\\/person\\\/552de1c88fa03b2180785f81655f8014\"},\"headline\":\"Kropki Cages Sudoku by Bill Murphy\",\"datePublished\":\"2026-04-20T07:00:48+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/\"},\"wordCount\":260,\"commentCount\":1,\"image\":{\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.gmpuzzles.com\\\/images\\\/puzzles2026\\\/260420-KropkiCagesSudoku.png\",\"keywords\":[\"sudoku\",\"wed\\\/thurs\",\"variation\",\"gas\"],\"articleSection\":[\"Puzzle\",\"Other Sudoku\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/\",\"url\":\"https:\\\/\\\/www.gmpuzzles.com\\\/blog\\\/2026\\\/04\\\/kropki-cages-sudoku-by-bill-murphy\\\/\",\"name\":\"Kropki Cages Sudoku by Bill Murphy - 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