HMMT November

Combinatorics Difficulty 5.0 AIME, harder Find the answer

On an $8 \times 8$ chessboard, 6 black rooks and $k$ white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The answer is $k=14$ . For a valid construction, place the black rooks on cells $(a, a)$ for $2 \leq a \leq 7$ and the white rooks on cells $(a, a+1)$ and $(a+1, a)$ for $1 \leq a \leq 7$ . Now, we prove the optimality. As rooks can only attack opposite color rooks, the color of rooks in each row is alternating. The difference between the number of black and white rooks is thus at most the number of rooks. Thus, $k \leq 6+8=14$ .

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