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Rubik's Cube Scrambling Requires at Least 26 Random Moves

Yanlin Qu, Tomas Rokicki, Hillary Yang

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.

Rubik's Cube Scrambling Requires at Least 26 Random Moves

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.

Paper Structure

This paper contains 10 sections, 1 theorem, 8 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rubik's Cube notation
  4. Markov chain mixing time
  5. Big Red 200
  6. Main result
  7. "Trilateration"
  8. Distance to the origin
  9. Distance to the superflip
  10. Distance to the checkerboard

Key Result

Theorem 1

Rubik’s Cube scrambling requires at least 26 random moves, i.e., $t_{\mathrm{mix}}\geq26.$

Figures (8)

  • Figure 1: U: up, D: down, F: front, B: back, L: left, R: right; 1: 90$^\circ$, 2: 180$^\circ$, 3: 270$^\circ$
  • Figure 2: Left: the origin $o$. Middle: the superflip $s$. Right: the checkerboard $c$.
  • Figure 3: The decay curve of $\mathrm{TV}(d_o(X_n),d_o(X_\infty))$.
  • Figure 4: The distributions of $d_o(X_{10})$, $d_o(X_{20})$, $d_o(X_{30})$, and $d_o(X_\infty)$.
  • Figure 5: The decay curve of $\mathrm{TV}(d_s(X_n),d_s(X_\infty))$.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem
  • proof