A king makes a tour of the chessboard, moving one square at a time
orthogonally or diagonally, visiting
each intermediate square exactly once, and returning to the starting square
on the last move.

Prove that the number of diagonal moves that the king made is even.

Since the king ended up on the same square, and visited each
intermediate square exactly once, the number of moves he made
was 64.

Also note that for each orthognal move, the colours of the starting
and ending squares are opposite.
Since the king arrives finishes at a square of the same colour, the
number of orthogonal moves must be even.

Hence the number of diagonal moves is 64 - an even number, which
must also be even.