How to solve the torus puzzle (Q1736499)

Summary: In this paper, we consider the following sliding puzzle called torus puzzle. In an \(m\) by \(n\) board, there are \(mn\) pieces numbered from 1 to \(mn\). Initially, the pieces are placed in ascending order. Then they are scrambled by rotating the rows and columns without the player's knowledge. The objective of the torus puzzle is to rearrange the pieces in ascending order by rotating the rows and columns. We provide a solution to this puzzle. In addition, we provide lower and upper bounds on the number of steps for solving the puzzle. Moreover, we consider a variant of the torus puzzle in which each piece is colored either black or white, and we present a hardness result for solving it.