Solution to the Number Rotation Puzzle

TL;DR

This paper analyzes the Number Rotation Puzzle, identifying solvability conditions and providing algorithms, including a novel construction related to the outer automorphism of S_6, for various board sizes and configurations.

Contribution

It offers a comprehensive analysis of the NRP's solvability, algorithms for solving configurations, and a novel mathematical construction related to S_6 automorphisms.

Findings

Solvability depends on parity restrictions for large boards.

Special conditions apply for smaller board and block sizes.

Introduces a new construction of the outer automorphism on S_6.

Abstract

The Number Rotation Puzzle (NRP) is a combination puzzle in which the goal is to rearrange a scrambled rectangular grid of numbers back into order via moves that consist of rotating square blocks of numbers of fixed size. Over all possible boards and rotating block sizes, we find all solvable initial configurations and provide algorithms to solve such configurations. For sufficiently large board and rotating block sizes, solvability conditions depend only on parity restrictions, with special additional conditions for smaller sizes. One special case leads to a novel construction of the exotic outer automorphism on $S_6$.