A king visited all squares of the usual $8 \times 8$ chessboard exactly once starting from the lower left square and finishing at the upper right square. Prove that the king made at least one diagonal move.

Let's argue indirectly...

Assume that the king made no diagonal moves.

Then, on each move the king must have changed the color of its square.

Now consider the list of all squares, in order, according to when they were visited by the king. This list starts with a white square and ends with a white square, with the colors of the squares alternating in between. Hence, there must be an odd number of squares.

But the king visited all 64 squares exactly once. This is our contradiction.

Hence, our assumption must be rejected, and its opposite must be true: the king must have made at least one diagonal move.