Question

一个边长为整数的正方形被切成 $10$ 个正方形，所有小正方形边长都是整数，并且至少 $8$ 个小正方形的面积为 $1$ 。原正方形边长的最小可能值是多少？

A square with an integer side length is cut into $10$ squares, all of which have integer side length and at least $8$ of which have area $1.$ What is the smallest possible value of the length of the side of the original square?

$3$

$4$

$5$

$6$

$7$

答案：B

知识点：铺砖极端原理

难度评级：1540

小提示：

边长为 $3$ 时，总面积太小，无法切成十个整数边长正方形。

A side length of $3$ gives area too small for ten integer squares.

大提示：

一个 $4$ 乘 $4$ 正方形可以按图示方式切分。

A $4$ by $4$ square can be cut as shown.

视频讲解：

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文字解答：

因为这 $10$ 个小正方形的边长都是正整数，所以每个小正方形的面积至少为 $1$ ，总面积至少为 $10$ 。边长不超过 $3$ 的整数边长正方形面积至多为 $3^2=9$ ，所以大正方形的边长不能小于 $4$ 。

下图把一个 $4\times4$ 正方形切成十个整数边长的正方形，其中八个是单位正方形。因此边长 $4$ 可以达到，而且是最小值。

所以正确答案是 B。

Since all $10$ squares have positive integer side lengths, each has area at least $1$ . Their total area is therefore at least $10$ . A square with integer side length at most $3$ has area at most $3^2=9$ , so its side length cannot be less than $4$ .

The following configuration cuts a $4\times4$ square into ten integer-sided squares, eight of which are unit squares. Therefore side length $4$ is attainable and is the minimum.

Thus, the answer is B .

2012 AMC 8 第 17 题