We classify an algebraic phenomenon on several families of wreath products that can be seen as coming from a generalization of a puzzle about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we build upon a paper of Bar Yehuda, Etzion, and Moran, a paper of Ehrenborg and Skinner, and a paper of Rabinovich to provide perhaps the fullest generalization yet, modeling both the switches and the spinning table as arbitrary finite groups combined via a wreath product. We classify large families of wreath products depending on whether or not they correspond to a solvable puzzle, completely classifying the puzzle in the case when the switches behave like abelian groups, constructing winning strategies for all wreath products that are p-groups, and providing novel examples for other puzzles where the switches behave like nonabelian groups, including the puzzle consisting of two interchangeable copies of the monster group M. Lastly, we provide a number of open questions and conjectures, and provide other suggestions of how to generalize some of these ideas further.

我们对几个花环产品系列的代数现象进行了分类，这些代数现象可以被视为来自旋转台角上的开关难题的概括。自从 Martin Gardner 在 1979 年首次普及此类谜题以来，就已经有人对它们进行了论述和推广。在本文中，我们以 Bar Yehuda、Etzion 和 Moran 的论文、Ehrenborg 和 Skinner 的论文以及 Rabinovich 的论文为基础，提供了迄今为止最全面的概括，将开关和旋转台建模为通过花环积组合的任意有限组。我们根据花环产品是否对应于可解决的谜题对大族进行分类，在开关表现得像阿贝尔群的情况下对谜题进行完全分类，为所有符合条件的花环产品构建获胜策略p组，并为其他谜题提供新颖的示例，其中开关的行为类似于非阿贝尔组，包括由怪物组M的两个可互换副本组成的谜题。最后，我们提供了一些悬而未决的问题和猜想，并提供了如何进一步推广其中一些想法的其他建议。