How to programmatically solve the 15 (moving numbers) puzzle?

Question

all of you have probably seen the moving number/picture puzzle. The one where you have numbers from 1 to 15 in a 4x4 grid, and are trying to get them from random starting positi

Answer 1

The solution strategy described by the original poster will always work for a standard solvable 15-puzzle. If Axarydax can reduce a 15-puzzle to the state s/he described and still be unable to solve it, then it was impossible to begin with. Let me explain.

If we treat the blank space in the puzzle as one of the tiles, then each legal move involves swapping that blank "tile" for an adjacent tile. This allows us to regard motions on the puzzle as permutations on 16 characters. That is, elements of the symmetric group S₁₆. Each primitive move is a "swap" or transposition between only two elements (one of which is the blank).

Because the puzzle begins and ends with the blank tile in the lower right, the blank tile must move an even number of times for the puzzle to be solved. (This is easiest to see by imagining an overlaid checkerboard pattern on top of the puzzle -- after an odd number of moves the blank would be on a different color square.) That means that the solution enacted must be a product of evenly many permutations, so it must be an element of the alternating group A₁₆, which has exactly half of S₁₆. (Of the 16! permutations of S₁₆, 16!/2 permutations are even, and 16!/2 are odd. Moreover even*even=even, even*odd = odd, and odd*odd=even.)

If the necessary correcting permutation happens to be odd, it's not possible to solve the puzzle, no matter what you do. If the necessary correcting permutation is even, and if Axarydax follows the strategy described, then the permutation required for the remaining 2x2 block will necessarily be an even permutation fixing the blank square. The only even permutations of only three elements are the rotations 1->2->3->1 (cycle notation (123)) and 1->3->2->1 (cycle notation (132)). These are easily performed on the remaining four squares without disturbing the others.

Since it's implausible that the Axarydax cannot figure out these trivial solutions of the 2x2 blocks, I suspect that either s/he has been pranked, or the 15-puzzle being attempted is nonstandard in some way.