Paper by Erik D. Demaine

Reference:

Erik D. Demaine and Mikhail Rudoy, “A simple proof that the (n² − 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 n² − 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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