A simple proof that the (n^2-1)-puzzle is hard
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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 Ratner and Warmuth (1990). We provide an alternative simpler proof of this fact by reduction from the rectilinear Steiner tree problem.
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