65 bugs are placed at different squares of a 9x9 square board. A bug in each move moves to a horizontal or vertical adjacent square. No bug makes two horizontal or two vertical moves in succession. Show that after some moves, there will be at least two bugs in the same square.

Using a checkerboard-style coloring of the 81 squares into 16 shaded and 65 unshaded (or similar structural split), a bug starting on a shaded square returns to some shaded square within at most 4 moves, and a bug starting on an unshaded square returns to a shaded square within at most 3 moves. Since there are only a limited number of shaded squares available at each relevant step but 65 bugs must eventually funnel through them, the pigeonhole principle forces at least two bugs into the same square after a bounded number of moves.