Solving infinitary Rubik's cubes

Abstract

We develop infinitary analogues of the $N\times N\times N$ Rubik's cube. We'll be pushed to consider the possibility of transfinitely many twists and the foremost question we shall study is whether or not all infinite scrambles are solvable, in principle, and in how many twists. As is typical of infinitary generalizations of everyday games and puzzles, several alternative definitions are reasonable, including in particular the edged and edgeless cubes, which bear surprising theoretical differences, not analogous to the finite case. We show that for the edged cube of cardinality $\aleph_\alpha$, all convergent (in a suitable sense) scrambles are in fact solvable in principle in fewer than $\omega_{\alpha+1}$ many moves. For the countable edgeless variation, we prove by entirely different methods that all convergent scrambles are solvable in a mere $\omega^2$ many moves and this solution does not require knowledge of how the scrambled configuration was obtained. Finally, we explore the space of all legal configurations of the countable edgeless cube connected to the solved configuration by accessibility. We invite several open questions, including the solvability in principle of edgeless cubes of uncountable cardinality.