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HMMT November

Combinatorics Difficulty 5.3 AIME, harder Prove it United States

Problem:

On an 8×88 \times 8 chessboard, 6 black rooks and kk white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of kk.

(Two rooks attack each other if they are in the same row or column and no rooks are between them.)

Solution

Solution:

The answer is k=14k=14. For a valid construction, place the black rooks on cells (a,a)(a, a) for 2≤a≤72 \leq a \leq 7 and the white rooks on cells (a,a+1)(a, a+1) and (a+1,a)(a+1, a) for 1≤a≤71 \leq a \leq 7.

Figure 1

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≤6+8=14k \leq 6+8=14.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.