AMC8 2024

AMC8 2024 · Q17

AMC8 2024 · Q17. It mainly tests Basic counting (rules of product/sum), Geometry misc.

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 $3$ x $3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3$ x $3$ grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?

国际象棋中的国王可以攻击与其相距一步的格子，包括水平、垂直或对角线方向。例如，一个国王放在 $3 \times 3$ 网格的中心格子，会攻击其他 $8$ 个格子，如下图所示。假设一个白国王和一个黑国王放在 $3 \times 3$ 网格的不同格子上，且它们互不攻击（也就是说，不紧挨着）。有几种放置方式？

(A) \ 20 \ 20

(B) \ 24 \ 24

(D) \ 28 \ 28

(E) \ 32 \ 32

Answer

Correct choice: (E)

正确答案：（E）

Solution

If you place a king in any of the $4$ corners, the other king will have $5$ spots to go and there are $4$ corners, so $5 \times 4=20$. If you place a king in any of the $4$ edges, the other king will have $3$ spots to go and there are $4$ edges so $3 \times 4=12$. That gives us $20+12=32$ spots for the other king to go into in total. So $\boxed{\textbf{(E)} 32}$ is the answer.

如果你把一个国王放在 $4$ 个角中的任意一个，另一个国王有 $5$ 个位置可放，有 $4$ 个角，所以 $5 \times 4=20$。如果你把一个国王放在 $4$ 个边中的任意一个，另一个国王有 $3$ 个位置可放，有 $4$ 个边所以 $3 \times 4=12$。总共 $20+12=32$ 个位置给另一个国王放。所以答案是 $\boxed{\textbf{(E)} 32}$。

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