{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/probabilistic-estimates-of-the-diameters-of","title":"Probabilistic estimates of the diameters of the Rubik's Cube groups","arxiv_id":"2404.07337","date":"2024-04-10","proceeding":null,"authors":["So Hirata"],"abstract":"The diameter of the Cayley graph of the Rubik's Cube group is the fewest number of turns needed to solve the Cube from the hardest initial configuration. For the 2$\\times$2$\\times$2 Cube, the diameter is 11 in the half-turn metric, 14 in the quarter-turn metric, 19 in the semi-quarter-turn metric, and 10 in the bi-quarter-turn metric. For the 3$\\times$3$\\times$3 Cube, the diameter was determined by Rokicki et al. to be 20 in the half-turn metric and 26 in the quarter-turn metric. This study shows that a modified version of the coupon collector's problem in probability theory can predict the diameters correctly for both 2$\\times$2$\\times$2 and 3$\\times$3$\\times$3 Cubes insofar as the quarter-turn metric is adopted. In the half-turn metric, the diameters are overestimated by one and two, respectively, for the 2$\\times$2$\\times$2 and 3$\\times$3$\\times$3 Cubes, whereas for the 2$\\times$2$\\times$2 Cube in the semi-quarter-turn and bi-quarter-turn metrics, they are overestimated by two and underestimated by one, respectively. Invoking the same probabilistic logic, the diameters of the 4$\\times$4$\\times$4 and 5$\\times$5$\\times$5 Cubes are predicted to be 48 (41) and 68 (58) in the quarter-turn (half-turn) metric, whose precise determinations are far beyond reach of classical supercomputing. The probabilistically estimated diameter is shown to obey the approximate formula of $\\ln N / \\ln r + \\ln N / r$, where $N$ is the number of configurations and $r$ is the branching ratio.","url_abs":"https://arxiv.org/abs/2404.07337v7","url_pdf":"https://arxiv.org/pdf/2404.07337v7.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"probabilistic-estimates-of-the-diameters-of","repo_url":"https://github.com/sohirata/rubik","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}