2025 AMC 12B 第 13 题

Question

一个圆被分成

6

个大小不同的扇形。接着把其中

2

个扇形涂红、

2

个涂绿、

2

个涂蓝，并要求相邻的两个扇形颜色不同。下图展示了一种涂色方式。

一共有多少种不同的涂色方式？

A circle has been divided into

6

sectors of different sizes. Then

2

of the sectors are painted red,

2

painted green, and

2

painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below.

How many different colorings are possible?

12

16

18

24

28

答案：D

知识点：图论乘法原理

难度评级：1660

小提示：

同色扇形不能相邻，所以每种颜色的两个扇形必须形成一对不相邻的位置。

Because same-colored sectors cannot touch, the two sectors of each color form a non-adjacent pair

大提示：

先数把

6

个扇形分成三对不相邻扇形的方法，再把三种颜色分配给这些对。

Count the ways to split the

6

sectors into three non-adjacent pairs, then assign the three colors to the pairs

解答：

每种颜色的两个扇形必须不相邻，所以一种涂色方案就是将

6

个环形排列的扇形分成三对不相邻的位置，再为三对分配颜色。不相邻的位置对应于

6

环的补图，该补图是三棱柱图，有

4

个完美匹配。再用

3!

种方式将三种颜色分配给三对，共有

4 \times 6 = 24

种涂色方案。

所以正确答案是 D。

The two sectors of each color must be a non-adjacent pair, so a coloring is a way to split the

6

cyclic sectors into three non-adjacent pairs together with an assignment of the three colors. The non-adjacent pairs are the edges of the complement of the

6

-cycle, the triangular prism, which has

4

perfect matchings. Assigning the three colors in

3!

ways gives

4 \times 6 = 24

colorings.

Thus, the correct answer is D.

2025 AMC 12B 第 13 题

答案：D

难度评级：1660

小提示：

大提示：

解答：

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