$25$ counters, numbered from $1$ to $25,$ are placed on the squares of a $5 \times 5$ chessboard, as shown below.
Mabel wants to move each counter to an adjacent square (horizontally or vertically, but not diagonally), so that as before, each square contains exactly one counter. In how many ways can they do this?
I'm pretty sure we need to use a one-to-one correspondence or the pigeonhole principle, but I don't know how to go about doing it.
After doing this operation, the $13$ counters on white tiles will each move to one of the $12$ black tiles. By Pigeonhole, a black tile will be used at least twice. So there are no ways to do this procedure.