How can I ensure that when I shuffle my puzzle I still end up with an even permutation?

Question

I\'m interested making an implementation of the 14-15 puzzle:

Answer 1

You should be able to use Fischer-Yates.

• Generate a random permutation using Fischer-Yates.
• Check if it is even.
• If it is not even, swap the first two elements of the permutation.

Consider an even permutation P = x1 x2 .... xn.

Fischer yates generates P with probabilty 1/n!.

It generates x2 x1 ... xn with probability 1/n!.

Thus the probability that the above process generates the permutation P is 2/n! = 1/(n!/2)

n!/2 is the number of even permutations.

Thus the above process generates even permutations with same probability.

To check if a permutation is even: count the parity of the number of inversions in the permutation.