What can be said about the difficulty of solving this kind of puzzle, in general? Martin Gardner devoted his February, 1964 Mathematical Games column to sliding-block puzzles. This is what he had
to say:These puzzles are very much in want of a theory. Short of trial and error, no one knows how to determine if a given state is obtainable from another given state, and if it is obtainable, no one knows how to find the minimum chain of moves for achieving the desired state.Forty years later, we still do not have such a theory. It turns out there is a good reason for this: sliding-block puzzles have recently been shown to belong to a class of problems known as PSPACE-complete. These problems are thought to be even harder than their better-known counterparts, the NP-complete problems (such as the traveling salesman problem).
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posted by psychotic_venom at 6:47 AM on June 22, 2004