AMC8 1991

AMC8 1991 · Q5

AMC8 1991 · Q5. It mainly tests Pigeonhole principle.

A "domino" is made up of two small squares: $\square$. Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?

一个“多米诺骨牌”由两个小方格组成：$\square\square$。下面所示的哪个“棋盘”无法被整数量目的不重叠的多米诺骨牌完全精确覆盖？

(A) $3 \times 4$ $3 \times 4$

(B) $3 \times 5$ $3 \times 5$

(D) $4 \times 5$ $4 \times 5$

(E) $6 \times 3$ $6 \times 3$

Answer

Correct choice: (B)

正确答案：（B）

Solution

A collection of non-overlapping dominoes must cover an even number of squares. Since checkerboard (B) has an odd number of squares, it follows that it cannot be covered as required. A little experimentation shows how the other checkerboards can be covered.

非重叠多米诺骨牌的集合必须覆盖偶数个方格。由于棋盘 (B) 有奇数个方格，因此无法按要求覆盖。稍作尝试即可看出其他棋盘如何覆盖。

Topics

CP.005 Pigeonhole principle