Solving infinitary Rubik's cubes

TL;DR

This paper extends the Rubik's cube to infinite and transfinite sizes, analyzing the solvability of various infinite variants and establishing bounds on the number of moves needed for solutions.

Contribution

It introduces infinitary analogues of the Rubik's cube, proves solvability bounds for these variants, and explores the structure of their configuration spaces.

Findings

All convergent scrambles of the edged cube of size 1_0 are solvable in fewer than 5_{0+1} moves.

All convergent scrambles of the countable edgeless cube are solvable in 5^2 moves without prior configuration knowledge.

The configuration space of the countable edgeless cube connected to the solved state is characterized and open questions are posed.

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.