HMMT 十一月 2021 · 冲刺赛 · 第 20 题

HMMT November 2021 — Guts Round — Problem 20

[11] On a chessboard, a queen attacks every square it can reach by moving from its current square along a row, column, or diagonal without passing through a different square that is occupied by a chess piece. Find the number of ways in which three indistinguishable queens can be placed on an 8 × 8 chess board so that each queen attacks both others.

解析

[11] On a chessboard, a queen attacks every square it can reach by moving from its current square along a row, column, or diagonal without passing through a different square that is occupied by a chess piece. Find the number of ways in which three indistinguishable queens can be placed on an 8 × 8 chess board so that each queen attacks both others. Proposed by: Gabriel Wu Answer: 864 Solution: Q Q Q Q Q Q The configuration of three cells must come in a 45-45-90 triangle. There are two cases, both shown above: the triangle has legs parallel to the axes, or it has its hypotenuse parallel to an axis. The first case can be solved by noticing that each selection of four cells in the shape of a square corresponds to 2 2 four such possibilities. There are 7 possible squares of size 2 × 2, 6 possible squares of size 3 × 3, and 2 2 2 so on. The total for this first case is thus 4(7 + 6 + · · · + 1 ) = 560. The second case can also be done by casework: each triangle in this case can be completed into an n + 1 by 2 n + 1 rectangle, of which there are 7 · 6 + 6 · 4 + 5 · 2 (for n = 1 , 2 , 3 respectively). Multiply this by 4 to get all orientations of the triangle. The final answer is 560 + 4(7 · 6 + 6 · 4 + 5 · 2) = 864.