Question

在下图所示的平面图形中，有 $3$ 个单位正方形已被涂色。至少还需要涂色多少个额外的单位正方形，才能使所得图形有两条对称轴？

In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

$4$

$5$

$6$

$7$

$8$

答案：D

知识点：对称性

难度评级：1270

小提示：

两条对称轴必须是这个矩形的中央竖直线和中央水平线

The two symmetry lines must be the central vertical and central horizontal lines of the rectangle

大提示：

每个已涂色的正方形都会迫使它关于两条轴的反射位置也被涂色

Each shaded square forces its reflections across both lines to be shaded too

解答：

要同时具有这两种对称性，对称轴必须是 $5$ 乘 $4$ 网格的竖直和水平中心线。每个已涂色正方形都会迫使它关于每条轴反射得到的正方形也被涂色。

顶部的正方形不在中心线上，所以它的反射组共有 $4$ 个正方形，需要再涂 $3$ 个。中间的正方形在中央列上，所以它的反射组共有 $2$ 个正方形，需要再涂 $1$ 个。右下的正方形同样有一个大小为 $4$ 的反射组，需要再涂 $3$ 个。

至少还需涂色的正方形数为 $3 + 1 + 3 = 7$ 。

因此，正确答案是 D。

For both symmetries, the lines must be the vertical and horizontal center lines of the $5$ -by- $4$ grid. Every shaded square then forces the squares obtained by reflecting it across each line.

The top square lies off-center, so its reflection group has $4$ squares, requiring $3$ more. The middle square sits on the central column, so its group has $2$ squares, requiring $1$ more. The bottom-right square again has a group of $4,$ requiring $3$ more.

The least number of additional squares is $3 + 1 + 3 = 7.$

Thus, D is the correct answer.

2020 AMC 12A 第 6 题

答案：D

难度评级：1270

小提示：

大提示：

解答：

其他年份的第 6 题