{"id":2360,"date":"2024-12-30T13:44:19","date_gmt":"2024-12-30T13:44:19","guid":{"rendered":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/?p=2360"},"modified":"2024-12-30T13:58:50","modified_gmt":"2024-12-30T13:58:50","slug":"8-questions-to-chatgpt-about-rubiks-cubes","status":"publish","type":"post","link":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/2024\/12\/30\/8-questions-to-chatgpt-about-rubiks-cubes\/","title":{"rendered":"Eight Questions to ChatGPT about Rubik\u2019s Cubes"},"content":{"rendered":"\n<p class=\"has-cyan-bluish-gray-background-color has-background\">1. What is the distance of a random starting configuration from a correct solution?<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">2. There\u2019s obviously a minimum distance: Presumably there is a maximum distance?&nbsp;<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">3. I guess a virtual cube could be simulated, with a virtual mouse, like a complex \u201cjoy-cube\u201d computationally configurable to a specified distance from a solution, and manipulable by players?<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">4. Real players could be tested for their speed (number of manipulations) at reaching a solution from (a) a given starting distance, (b) an average starting distance, or (c) the maximal starting distance?&nbsp;<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">5. They could also be trained and tested from differently sized and shaped virtual&nbsp;&nbsp;\u201ccubes\u201d, all the way up to a (digitally approximated) virtual \u201csphere\u201d?<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">6. Besides testing real players with simulations on a mouse, I suppose algorithms could be generated and tested (and trained) for their \u201cspeed\u201d (in number of manipulations) to reach a solution?<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">7. These are all parameters. Do they have any meaning or use (analogous to the way&nbsp;&nbsp;the parameters of prime number distributions are studied by number theorists, when they are trying to generating potential proofs or theorems)?&nbsp;<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-background-color has-background\">8. Or am I talking nonsense (or trivia)?<\/p>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-ea3083c6ff01fe9bd3375d6e29b0b2cf\"><strong>1. What is the distance of a random starting configuration from a correct solution?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Answer Overview:<\/strong><br>The \u201cdistance\u201d typically refers to how many moves are required, under a given move metric, to transform a scrambled cube into its solved state. On a standard 3\u00d73\u00d73 Rubik\u2019s Cube, this is often measured in either:\n<ol start=\"1\" class=\"wp-block-list\">\n<li><strong>Quarter-Turn Metric (QTM)<\/strong>\u00a0\u2013 each 90\u00b0 turn counts as 1 move.<\/li>\n\n\n\n<li><strong>Half-Turn Metric (HTM)<\/strong>\u00a0\u2013 a 180\u00b0 turn counts as 1 move, and a 90\u00b0 turn also counts as 1 move.<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li><strong>Average Distance for a Random Configuration:<\/strong><br>Most random configurations lie near (but not exactly at) the upper part of the possible \u201cdistance\u201d spectrum. Empirical research suggests that, in half-turn metric, many random scrambles take around 17\u201318 moves (on average) to solve optimally. In quarter-turn metric, that average is a bit higher (19\u201320 moves).<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-67618954e0f23d0392c29efa92cf339d\"><strong>2. There\u2019s obviously a minimum distance: Presumably there is a maximum distance?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Minimum Distance:<\/strong><br>The minimum distance is 0 moves if you\u2019re already in the solved state.<\/li>\n\n\n\n<li><strong>Maximum Distance:<\/strong>\n<ul class=\"wp-block-list\">\n<li>In quarter-turn metric (QTM), the proven \u201cGod\u2019s number\u201d for the 3\u00d73\u00d73 cube is 26. This means\u00a0<em>no<\/em>configuration requires more than 26 quarter-turn moves to solve.<\/li>\n\n\n\n<li>In half-turn metric (HTM), the proven number is 20. No configuration requires more than 20 half-turn moves.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Interpretation:<\/strong><br>This maximum distance is the farthest any scramble can be from the solved configuration under optimal solving.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-e49376c89968bde054cec622e6691f15\"><strong>3. I guess a virtual cube could be simulated, with a virtual mouse, like a complex \u201cjoy-cube\u201d computationally configurable to a specified distance from a solution, and manipulable by players?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Implementation Feasibility:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Virtual Mouse \/ \u201cJoy-cube\u201d:<\/strong>\u00a0It is indeed possible to create a 3D interactive interface (e.g., with WebGL or a game engine) that:\n<ol start=\"1\" class=\"wp-block-list\">\n<li>Generates a specific cube scramble with a known \u201cdistance\u201d from solved.<\/li>\n\n\n\n<li>Allows users to perform standard legal moves with a mouse or other input device.<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li><strong>Distance Configuration:<\/strong><br>Generating scrambles at a precise known distance is more challenging because you need to work backward from the solved state using a known sequence length. However, it can be done by applying exactly\u00a0ddrandom moves from the solved state to ensure the scramble is at distance\u00a0dd\u00a0or fewer.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Practical Applications:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Training &amp; Education:<\/strong>\u00a0Players can practice on configurations that systematically increase in difficulty (i.e., distance).<\/li>\n\n\n\n<li><strong>Competitive Settings:<\/strong>\u00a0Timed challenges can be set up where each puzzle is guaranteed to be at the same distance from solved.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-4940bd6372ea1fa62b8f1489e744fa30\"><strong>4. Real players could be tested for their speed (number of manipulations) at reaching a solution from (a) a given starting distance, (b) an average starting distance, or (c) the maximal starting distance?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Yes, this is a viable testing methodology.<\/strong>\n<ol start=\"1\" class=\"wp-block-list\">\n<li><strong>(a) Given Starting Distance:<\/strong>\n<ul class=\"wp-block-list\">\n<li>A known scramble distance tests how players adapt their solving strategies or heuristics when they are certain how \u201cdeep\u201d the puzzle is from solved.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>(b) Average Starting Distance:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Random scrambles reflect real-world solves. This is how most speedcubers test their times in competitions.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>(c) Maximal Starting Distance:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Testing how long it takes a solver to unravel the worst-case scenario can be used to measure both skill and stamina (in solving). This is less common in real competitions but interesting for theoretical or research purposes.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-8c4fc44a76facb87bd75e5a9043b3be7\"><strong>5. They could also be trained and tested from differently sized and shaped virtual \u201ccubes\u201d, all the way up to a (digitally approximated) virtual \u201csphere\u201d?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Different Puzzle Variants:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>NxN Cubes:<\/strong>\u00a0There are official NxN cubes (4\u00d74, 5\u00d75, etc.), and each has its own complexity and group theory nuances.<\/li>\n\n\n\n<li><strong>Other Twist Puzzles:<\/strong>\u00a0Megaminx, Pyraminx, Skewb, and more can also be simulated in 3D environments.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Virtual \u201cSphere\u201d:<\/strong>\n<ul class=\"wp-block-list\">\n<li>In puzzle terms, a sphere might be visualized as a puzzle with rotating \u201cbands.\u201d Some puzzle inventors have indeed created spherical twisty puzzles.<\/li>\n\n\n\n<li><strong>Why Train on Different Shapes?:<\/strong>\n<ol start=\"1\" class=\"wp-block-list\">\n<li>Increases familiarity with more generalized \u201cgroup moves.\u201d<\/li>\n\n\n\n<li>Offers variety in problem-solving approaches.<\/li>\n\n\n\n<li>Expands the solver\u2019s spatial reasoning.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-b0cf60c1e0f0192a391b831b92afaa50\"><strong>6. Besides testing real players with simulations on a mouse, I suppose algorithms could be generated and tested (and trained) for their \u201cspeed\u201d (in number of manipulations) to reach a solution?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Algorithmic Solvers:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Examples:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Kociemba\u2019s Algorithm for 3\u00d73\u00d73.<\/li>\n\n\n\n<li>Thistlethwaite\u2019s Algorithm (an older but historically significant approach).<\/li>\n\n\n\n<li>More recent optimizations in two-phase solvers.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Metrics &amp; Training:<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Move Count:<\/strong>\u00a0Minimizing the total number of moves to solve.<\/li>\n\n\n\n<li><strong>Algorithmic Complexity:<\/strong>\u00a0Some methods might produce short solutions but take longer in computation. Others might solve quickly (in real-time) but produce longer solutions.<\/li>\n\n\n\n<li><strong>Machine Learning Angle:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Reinforcement learning or other data-driven approaches can be tested at different distances, puzzle sizes, or puzzle shapes.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-b1295099791472b06ea0ac6b57395f4b\"><strong>7. These are all parameters. Do they have any meaning or use (analogous to the way the parameters of prime number distributions are studied by number theorists, when they are trying to generate potential proofs or theorems)?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Yes, they do have theoretical and practical meaning.<\/strong>\n<ol start=\"1\" class=\"wp-block-list\">\n<li><strong>Group Theory Significance:<\/strong>\n<ul class=\"wp-block-list\">\n<li>The Rubik\u2019s Cube (and related puzzles) are finite groups. Studying distances and configuration spaces informs research on group diameter, symmetry, and combinatorial aspects.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Algorithmic Complexity:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Identifying typical or worst-case puzzle states helps to explore \u201cGod\u2019s algorithm\u201d (optimal solutions) and the computational complexity behind them.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Heuristic &amp; Practical Value:<\/strong>\n<ul class=\"wp-block-list\">\n<li>In puzzle design or AI solvers, these parameters matter for how efficiently or quickly a puzzle can be solved.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Recreational Math &amp; Education:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Insights from these parameters help create teaching tools that illustrate concepts in math and computer science.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color has-medium-font-size wp-elements-510394fa507defcbae59e297a14a4645\"><strong>8. Or am I talking nonsense (or trivia)?<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>These questions are quite relevant to both recreational mathematics and more formal studies in group theory, search algorithms, and computational complexity. While these topics can seem niche, they have parallels to important problems in computer science (search algorithms and optimization), mathematics (group theory and combinatorics), and educational technology.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p><strong>Summary<\/strong><\/p>\n\n\n\n<p>Your questions touch on well-researched territory in Rubik\u2019s Cube mathematics and puzzle theory. Distances, maximum depths, testing methodologies, and algorithmic approaches all have real-world and theoretical applications. Far from being trivial, these areas intersect with math, computer science, education, and even psychology (e.g., how humans approach problem-solving).<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-content\/uploads\/sites\/287\/2024\/12\/image-1024x768.png\" alt=\"\" class=\"wp-image-2362\" srcset=\"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-content\/uploads\/sites\/287\/2024\/12\/image-1024x768.png 1024w, https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-content\/uploads\/sites\/287\/2024\/12\/image-300x225.png 300w, https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-content\/uploads\/sites\/287\/2024\/12\/image-768x576.png 768w, https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-content\/uploads\/sites\/287\/2024\/12\/image.png 1280w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>1. What is the distance of a random starting configuration from a correct solution? 2. There\u2019s obviously a minimum distance: Presumably there is a maximum distance?&nbsp; 3. I guess a virtual cube could be simulated, with a virtual mouse, like a complex \u201cjoy-cube\u201d computationally configurable to a specified distance from a solution, and manipulable by &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/2024\/12\/30\/8-questions-to-chatgpt-about-rubiks-cubes\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Eight Questions to ChatGPT about Rubik\u2019s Cubes&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3074,"featured_media":0,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[110,111,106,188],"tags":[],"class_list":["post-2360","post","type-post","status-publish","format-standard","hentry","category-cognitive-science","category-computation","category-language","category-rubiks-cube"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/posts\/2360","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/users\/3074"}],"replies":[{"embeddable":true,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/comments?post=2360"}],"version-history":[{"count":2,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/posts\/2360\/revisions"}],"predecessor-version":[{"id":2364,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/posts\/2360\/revisions\/2364"}],"wp:attachment":[{"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/media?parent=2360"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/categories?post=2360"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/generic.wordpress.soton.ac.uk\/skywritings\/wp-json\/wp\/v2\/tags?post=2360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}