Rubik's Cube Scrambling Requires at Least 26 Random Moves

TL;DR

This paper establishes a lower bound of 26 moves for effectively scrambling a standard Rubik's Cube, using Markov chain analysis and supercomputing to derive the first non-trivial bound on its mixing time.

Contribution

It introduces the first non-trivial lower bound on the number of random moves needed to thoroughly scramble a Rubik's Cube, using advanced computational methods.

Findings

Mixing time is at least 26 moves.

The analysis models the scramble as a Markov chain.

Provides the first non-trivial bound on Rubik's Cube scrambling complexity.

Abstract

Scrambling the standard 3x3x3 Rubik's Cube corresponds to a random walk on a group containing approximately 43 quintillion elements. Viewing the random walk as a Markov chain, its mixing time determines the number of random moves required to sufficiently scramble a solved cube. With the aid of a supercomputer, we show that the mixing time is at least 26, providing the first non-trivial bound.