Track / Stage 5 / 326 of 400 #926 of 1964

Problem 926

AIME late

Combinatorics Difficulty 5.8 Prove it

9. A chessboard is covered with 32 dominoes (each domino covers exactly two squares). Prove that these dominoes can be rotated by 90 or 180 degrees (each around the center of one of the squares it covers, they can be rotated independently and in any direction) so that the entire board is still covered.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

- Divide the board into $2 \times 2$ cells. Place the second layer of dominoes as follows: if a cell is entirely covered by a horizontal domino, place two vertical dominoes on it; otherwise, place two horizontal dominoes. As a result, no upper domino will coincide in position with the lower one. Now, let's recall the chessboard coloring and rotate each lower domino so that it occupies the position of the upper domino covering the same white cell.

(A. Shapovalov)

## Fourth Round

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.