Commutators in the Rubik’s Cube Group

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-10-20 DOI:10.1080/00029890.2023.2263158

Timothy Sun

引用次数: 0

Abstract

Since the Rubik’s Cube was introduced in the 1970s, mathematicians and puzzle enthusiasts have studied the Rubik’s Cube group, i.e., the group of all ≈4.3×1019 solvable positions of the Rubik’s Cube. Group-theoretic ideas have found their way into practical methods for solving the Rubik’s Cube, and perhaps the most notable of these is the commutator. It is well-known that the commutator subgroup of the Rubik’s Cube group has index 2 and consists of the positions reachable by an even number of quarter turns. A longstanding open problem, first posed in 2004, asks whether every element of the commutator subgroup is itself a commutator. We answer this in the affirmative and sketch a generalization to the n×n×n Rubik’s Cube, for all n≥2.

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魔方组中的换向子

自20世纪70年代魔方问世以来，数学家和解谜爱好者就开始研究魔方群，即魔方中所有≈4.3×1019可解位置的群。群论思想已经在解决魔方的实际方法中找到了自己的方式，其中最引人注目的也许是换向子。众所周知，魔方群的换向子群索引为2，由偶数个四分之一转所能到达的位置组成。2004年首次提出了一个长期存在的开放问题，即是否换向子子群的每个元素本身都是换向子。我们对这个问题的回答是肯定的，并对所有n≥2的情况下n×n×n魔方进行了概括。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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