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：

其他年份的第 13 题