Question

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?

$3 \times 4$

$3 \times 5$

$4 \times 4$

$4 \times 5$

$6 \times 3$

答案：B

知识点：铺砖奇偶性

难度评级：800

解答：

每个多米诺骨牌覆盖 $2$ 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： $3 \times 4 = 12$ ， $3 \times 5 = 15$ ， $4 \times 4 = 16$ ， $4 \times 5 = 20$ ， $6 \times 3 = 18$ 。只有 $3 \times 5 = 15$ 是奇数，所以不能被完全覆盖。

所以正确答案是 B。

Every domino covers exactly $2$ squares, so any board that is completely covered by non-overlapping dominoes must contain an even number of small squares.

Counting squares: $3 \times 4 = 12,$ $3 \times 5 = 15,$ $4 \times 4 = 16,$ $4 \times 5 = 20,$ and $6 \times 3 = 18.$ Only $3 \times 5 = 15$ 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 .

1991 AMC 8 第 5 题

答案：B

难度评级：800

解答：

其他年份的第 5 题