Question

你正在玩一个游戏。一个 $2\times 1$ 的长方形盖住了一个 $3\times 3$ 方格表中两个相邻的小正方形（横放或竖放都可能），但你不知道它盖住了哪两个小正方形。你的目标是找到至少一个被长方形盖住的小正方形。一次“回合”是你猜一个小正方形，然后会被告知这个小正方形是否被隐藏的长方形盖住。为了保证你猜过的小正方形中至少有一个被长方形盖住，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：1350

小提示：

想一想哪些小正方形可以不猜，同时仍然不可能完全避开这个骨牌

Think about which squares you could leave un-guessed without ever touching the domino

大提示：

未猜的小正方形中不能有两个相邻；在 $3\times 3$ 方格中，这样的集合最大有 $5$ 个小正方形

The un-guessed squares must contain no two adjacent squares; the largest such set in a $3\times 3$ grid has $5$ squares

解答：

一组猜过的小正方形能保证碰到这个骨牌，当且仅当未猜的小正方形中没有两个相邻；否则骨牌可以藏在那一对相邻小正方形上。 $3\times 3$ 方格中两两不相邻的小正方形集合最大是 $5$ 个小正方形的棋盘格集合（四个角加中心）。所以最多能留下 $5$ 个小正方形不猜，必须猜 $9-5=4$ 。

因此，正确答案是 C。

A set of guessed squares is guaranteed to hit the domino if and only if the un-guessed squares contain no two adjacent squares, since otherwise the domino could hide on that adjacent pair. The largest set of pairwise non-adjacent squares in the $3\times 3$ grid is the $5$ -square checkerboard (four corners plus the center). So at most $5$ squares can be left unguessed, and you must guess $9-5=4.$

Thus, the correct answer is C.

2023 AMC 12B 第 5 题

答案：C

难度评级：1350

小提示：

大提示：

解答：

其他年份的第 5 题