**Reference**:- Erik D. Demaine and Mikhail Rudoy, “A simple proof that the (
*n*^{2}− 1)-puzzle is hard”,*Theoretical Computer Science*, volume 732, July 2018, pages 80–84. **Abstract**:-
The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely
labeled unit squares within a 4 × 4 board in which the goal is
to slide the squares (without ever overlapping) into a target configuration.
By generalizing the puzzle to an
*n*×*n*board with*n*^{2}− 1 squares, we can study the computational complexity of problems related to the puzzle; in particular, we consider the problem of determining whether a given end configuration can be reached from a given start configuration via at most a given number of moves. This problem was shown NP-complete in [1]. We provide an alternative simpler proof of this fact by reduction from the rectilinear Steiner tree problem. **Comments**:- This paper is also available from ScienceDirect and as arXiv:1707.03146.
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Last updated August 19, 2022 by Erik Demaine.