Solution to the Number Rotation Puzzle

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 $S_6$.