1991 AMC 8 第 5 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
5.
一个“多米诺骨牌”由两个小正方形组成：
。下面哪一个“棋盘”不能被若干个不重叠的完整多米诺骨牌恰好完全覆盖？
A “domino” is made up of two small squares:
. Which of the “checkerboards” illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?
小提示：
每个多米诺骨牌恰好覆盖两个小正方形，所以先看每个棋盘有多少个小正方形
Each domino covers exactly two small squares, so focus on how many small squares each board has.
大提示：
棋盘要被完全覆盖，总格数必须是偶数；用两个维度相乘求每个棋盘的格数
A board can be fully covered only if its total number of squares is even; find the number of squares in each board by multiplying its two dimensions.
解答：
每个多米诺骨牌覆盖 个小正方形，所以能被不重叠多米诺完全覆盖的棋盘必须有偶数个小正方形。
各棋盘的格数为：，，，，。只有 是奇数，所以不能被完全覆盖。（每个格数为偶数的棋盘都有一条边长是偶数，很容易用多米诺骨牌铺满。）
所以正确答案是 B。
Every domino covers exactly squares, so any board that is completely covered by non-overlapping dominoes must contain an even number of small squares.
Counting squares: and Only is odd, so that board cannot be covered. (Each of the even boards has a side of even length and is easily tiled with dominoes.)
Thus, the correct answer is B .
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