Question

国际象棋中的王会攻击水平、竖直或对角方向相邻一步的所有方格。例如，在 $3 \times 3$ 网格中心方格上的王会攻击其他 $8$ 个方格，如下图所示。若一个白王和一个黑王放在 $3 \times 3$ 网格的不同方格上，并且它们互不攻击，有多少种放法？

A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3 \times 3$ grid so that they do not attack each other. In how many ways can this be done?

$20$

$24$

$27$

$28$

$32$

答案：E

知识点：分类讨论基本计数

难度评级：1410

解答：

先选白王所在方格。若白王在中心，它攻击所有其他方格，黑王有 $0$ 个可放位置。

若白王在角落，它攻击 $3$ 个方格，黑王有 $9-1-3=5$ 个安全格。角落有 $4$ 个，所以共有 $4\cdot5=20$ 种。

若白王在边上但不是角落，它攻击 $5$ 个方格，黑王有 $9-1-5=3$ 个安全格。这样的边中格有 $4$ 个，所以共有 $4\cdot3=12$ 种。

总数为 $20+12=32$ 。

所以正确答案是 E。

Count ordered placements by first choosing the square for the white king. If the white king is in the center, it attacks every other square, so there are $0$ choices for the black king.

If the white king is in a corner, it attacks $3$ squares, so the black king has $9-1-3=5$ safe squares. There are $4$ corner choices, giving $4\cdot5=20$ placements.

If the white king is on an edge but not a corner, it attacks $5$ squares, so the black king has $9-1-5=3$ safe squares. There are $4$ such edge choices, giving $4\cdot3=12$ placements.

The total number of placements is $20+12=32$ .

Thus, E is the correct answer.

2024 AMC 8 第 17 题

答案：E

难度评级：1410

解答：

其他年份的第 17 题