From the author’s abstract: “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 M. Gardner [Sci. Am. 240, No. 2, 16–27 (1979; doi:10.1038/scientificamerican0279-16)]. In this paper, we build upon a paper of R. Bar Yehuda et al. [Theor. Comput. Sci. 108, No. 2, 311–329 (1993; Zbl 0778.90090)], a paper of R. Ehrenborg and C. M. Skinner [J. Comb. Theory, Ser. A 70, No. 2, 249–266 (1995; Zbl 0833.90127)], and a paper of Y. Rabinovich [Inf. Process. Lett. 176, Article ID 106233, 8 p. (2022; Zbl 1486.91021)] 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 \(M\).” The last section of the paper provides further generalizations, and contains dozens of conjectures, open questions, and further directions of research.