Question

你正在玩一个游戏。一个 $2 \times 1$ 矩形覆盖了一个 $3 \times 3$ 方格网中两个相邻的小方格，方向可以水平或竖直，但你不知道它覆盖了哪两个小方格。你的目标是找到至少一个被矩形覆盖的小方格。一次“回合”包括你猜一个小方格，随后你会被告知该小方格是否被隐藏矩形覆盖。为了保证至少有一个猜中的小方格，你最少需要多少回合？

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

$3$

$5$

$4$

$8$

$6$

答案：C

知识点：组合游戏极端原理

难度评级：1560

解答：

假设每次猜测都没有命中。那么多米诺完全位于未猜的小方格中，所以未猜的小方格中必须含有两个相邻格。要保证命中，需要让未猜格中没有任何两个相邻。把方格黑白相间地染色。两两不相邻的小方格最多有 $5$ 个，即四个角和中心。因此至少要猜 $9 - 5 = 4$ 个格子。 $4$ 个也足够：猜四个边中点。每个隐藏矩形都会覆盖一个边中点，所以必定命中。正确答案是 C。

Suppose every guess misses. Then the domino lies entirely on unguessed squares, so those squares include two adjacent cells. To force a hit, we need the unguessed squares to have no two adjacent. Color the grid like a checkerboard. The biggest set of pairwise non-adjacent squares has $5$ cells, one color's worth: the four corners and the center. So we must guess at least $9 - 5 = 4$ squares. And $4$ is enough: guess the four edge midpoints, since every domino covers one square of each color, hence one edge midpoint. Therefore, the answer is C.

2023 AMC 10B 第 10 题

答案：C

难度评级：1560

解答：

其他年份的第 10 题