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Imagine a 5 by 5 grid, with numbers 1 to 24. One slot is left empty.

Now picture a 15 puzzle style game, where you have to set it up 1 - 24, then 25 magically appears.

As we all know, the 15 puzzle is not always possible, but how many cases of a 24 puzzle are impossible?

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The same argument for the $15$ puzzle applies. You need the starting position to be an even permutation, so $1/2$ of the starting positions are solvable. You follow the same proof that shows you can swap two pairs of numbers, but not one pair.

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