Solving infinitary Rubik's cubes

TL;DR

This work extends Rubik's cube ideas to infinitary settings by defining edged and edgeless infinite cubes and analyzing solvability under transfinitely many twists. It introduces convergence notions and cluster-based algebraic structures, proving that the edged cube is solvable in principle for any infinite cardinality with a bound $omega_{alpha+1}$ on the required moves, while the countable edgeless cube admits a constructive solution in at most $omega^2$ moves. The methods center on universally convergent twist sequences, twist-finite analysis, and cluster-move strategies that parallel finite-cube algorithms (notably adapting Demaine 2011), highlighting a sharp contrast between edge/corner constraints and the edgeless case. The results illuminate how infinite generalizations depend crucially on structural variants and open questions about uncountable cases and uniform solvers.

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 $\aleph_α$, all convergent (in a suitable sense) scrambles are in fact solvable in principle in fewer than $ω_{α+1}$ many moves. For the countable edgeless variation, we prove by entirely different methods that all convergent scrambles are solvable in a mere $ω^2$ 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.

Figures (4)

• Figure 1: Diagram indicating the alternate notations $R_\alpha = T_{x,\alpha}^{-1}$, $F_\beta = T_{z,\beta}^{-1}$, $U_\gamma = T_{y,\gamma}^{-1}$ for $\alpha,\beta,\gamma \ge 0$ and $R_\alpha', F_\beta', U_\gamma'$ are their respective inverses.
• Figure 2: Schematic of the initial stages of the solve in the white face. In each case, the pattern is understood to continue in the obvious way. Gray cells represent arbitrary colors here. Each stage $n$a involves finitely many highly parallelized cluster solution subroutines for $X_h\times\{n\}$ clusters (applications of Lemma \ref{['lemma:cluster_product_soln']}). Stage $n$b is analogous for $\{n\}\times Y_h$ clusters. Stage $n$c solves the individual diagonal $(n,n)$ cluster. The red outlined box indicated the region invariant through stage $n$ (and forever after). The meat of the proof is in the existence of these twist-finite sequences which solve infinitely many clusters at once and doing so without ever upsetting the increasing core of cells at the center of each face.
• Figure 3: Space of all legal configurations of the countable edgeless cube $\mathcal{Q}_{\aleph_0}$ connected to $f_\text{\upshape solved}$ by one- or two-way accessibility. The heavy border around the standard configurations is meant to suggest that there can be no other arrows in or out (except to illegal configurations). The dashed boundary between $\mathfrak L[f_\text{\upshape solved}]$ and $\mathfrak S$ is meant to suggest that this inclusion may or may not be proper.
• Figure 4: Configuration encoding a well-order $\preceq$ of $L$. On the left, the configuration after only $T_{x,\alpha},T_{y,\alpha},T_{x,\beta},T_{y,\beta},T_{x,\gamma},T_{y,\gamma}$ (because $\alpha \prec \beta \prec \gamma$) with the cross cells highlighted. On the right, (a piece of) the final configuration with the same cells highlighted to illustrate that they are unaffected by later twists in the sequence.

Theorems & Definitions (38)

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