Question

在一个 $3 \times 3$ 网格的每个小方格中放入一个 $\bigtriangleup$ 或 $\bigcirc$ 。下面显示了一个有三个 $\bigtriangleup$ 连成一线的示例。

有多少种配置同时有三个 $\bigtriangleup$ 连成一线，并且有三个 $\bigcirc$ 连成一线？

A $\bigtriangleup$ or $\bigcirc$ is placed in each of the nine squares in a $3 \times 3$ grid. Shown below is a sample configuration with three $\bigtriangleup$ 's in a line.

How many configurations will have three $\bigtriangleup$ 's in a line and three $\bigcirc$ 's in a line?

$39$

$42$

$78$

$84$

$96$

答案：D

知识点：分类讨论对称性

难度评级：1670

解答：

设三角形个数为 $k$ 。要同时有三角形线和圆形线，必须有 $3\le k\le6$ 。

若 $k=3$ ，三角形必须成一条线。行或列共有 $6$ 条可行；对角线不行，因为剩下的 $6$ 个圆形中没有完整圆形线。所以 $k=3$ 时有这些配置。由对称性， $k=6$ 也有六种。

若 $k=4$ ，先选一条三角形线。若这条线是行或列，额外的三角形有 $6$ 个位置可选，共 $6\cdot6=36$ 种；若这条线是对角线，则不能留下圆形线。由对称性， $k=5$ 也有 $36$ 种。

总数为 $6+36+36+6=84$ 。

正确答案是 D。

Let $k$ be the number of triangles. To have both a triangle line and a circle line, we need $3\le k\le6$ .

If $k=3$ , the three triangles must form a line. A row or column works, because the remaining six circles contain a full circle row or column. A diagonal does not work, because its complement contains no full line of circles. Thus there are $6$ configurations for $k=3$ . By symmetry, there are also $6$ configurations for $k=6$ .

If $k=4$ , choose the triangle line and then the extra triangle. If the line is a row or column, all $6$ possible extra positions work, because one parallel row or column remains all circles. This gives $6\cdot6=36$ configurations. If the triangle line is a diagonal, no extra position leaves a full circle line. By symmetry, $k=5$ also gives $36$ configurations.

The total number of configurations is $6+36+36+6=84$ .

Thus, the correct answer is D.

2022 AMC 8 第 23 题

答案：D

难度评级：1670

解答：

其他年份的第 23 题