AMC 8 2024 Problem 17
AMC8 2024 年第 17 题
排列与组合★★★☆☆
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 × 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 × 3 grid so that they do not attack each other. In how many ways can this be done?
在国际象棋中，"国王"可以进攻横向、纵向或对角线方向上所有距离一步的棋格。如下图所示，在一个3×3棋盘中，位于中心位置的"国王"可以进攻其他8个棋格。假设要在一个3×3棋盘的不同方格中放入一枚白色"国王"和一枚黑色"国王"，请问有多少种放置方式可以使得这两个"国王"不会互相进攻？
- A.20
- B.24
- C.27
- D.28
- E.32
正确答案 · Correct Answer
E
解析 · Solution
解法1
如果你将王放在任意一个角落，另一个王有5个位置可走，共有4个角落，所以5×4=20。如果你将王放在任意一条边上，另一个王有3个位置可走，共有4条边，所以3×4=12。总共得到20+12=32个位置供另一个王放置。因此答案是32。
我们看到中心对于任意一个国王来说都不是一个可行位置，因为它会攻击所有相邻的格子。这样就有三种组合：
角-角：共有4个角，它们之间没有正交或对角相邻，所以是
角-边：对于每个角，有两条边与之不相邻，
边-边：这种组合中唯一可行的是上边和左边，所以此类有2种
6+8+2=16
我们发现中心对国王来说是不行的。此外，国王不能彼此相邻。国王相邻的摆放方式有24种：16种角排列和8种边排列。不考虑条件时，共有8P2=56种摆放国王的方式。因此，我们的答案是56-24=32。
Solution 1
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 5×4=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 3×4=12. That gives us 20+12=32 spots for the other king to go into in total. So 32 is the answer., captaindiamond868 (minor edits)
We see that the center is not a viable spot for either of the kings to be in, as it would attack all nearby squares. This gives three combinations: Corner-corner: There are 4 corners, and none of them are touching orthogonally or diagonally, so it's Corner-edge: For each corner, there are two edges that don't border it, Edge-edge: The only possible combinations of this that work are top and left edges, so 2 for this type 6+8+2=16 We see that the center is off for the kings. Additionally, the kings cannot be next to each other. There are 24 ways for the kings to be placed adjacently: 16 corner permutations and 8 edge permutations. In total, there are 8P2=56 ways to arrange the kings conditions aside. Therefore, our answer is 56-24=32
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