# Solving the Rubik's Cube Optimally is NP-complete

## TL;DR

This paper proves that finding the optimal solution for an n x n x n Rubik's Cube is NP-complete, establishing computational complexity for this classic puzzle and its variants.

## Contribution

It introduces the first proof that solving an n x n x n Rubik's Cube optimally is NP-complete, using reductions from the Hamiltonian Cycle problem.

## Key findings

- Optimal Rubik's Cube solving is NP-complete.
- The proof applies to both Rubik's Square and standard Cube.
- Complexity results extend previous work on puzzle difficulty.

## Abstract

In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an $n \times n \times n$ Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square---an $n \times n \times 1$ generalization of the Rubik's Cube---and then proceed with a similar but more complicated proof for the Rubik's Cube case.