1990 AMC 12 第 9 题

Question

一个立方体的每条棱都被涂成红色或黑色。立方体的每个面都至少有一条黑棱。黑棱数的最小可能值是

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is

2

3

4

5

6

答案：B

知识点：cube covering argument construction

难度评级：1450

小提示：

每条棱恰好属于两个面

Each edge belongs to exactly two faces

大提示：

为构造达到下界的情形，选择三条互不相邻的棱

For the matching upper bound, choose three mutually nonadjacent edges

解答：

每条黑棱只能覆盖与它相邻的两个面，所以要覆盖全部

6

个面，至少需要

3

条黑棱。选择分别属于“顶面与前面”“底面与左面”“后面与右面”这三对面的棱。这三条棱覆盖全部六个面，所以最小值为

3

。

所以正确答案是 B。

Each black edge can cover only its two incident faces, so covering all

6

faces requires at least

3

black edges. Choose edges incident to the face-pairs top/front, bottom/left, and back/right. These three edges cover all six faces, so the minimum is

3.

Thus the correct answer is B.

1990 AMC 12 第 9 题

答案：B

难度评级：1450

小提示：

大提示：

解答：

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