Question

一个 $11\times11\times11$ 的木质立方体由 $11^3$ 个单位立方体粘成。从一个观察点最多能看到多少个单位立方体？

An $11\times11\times11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

$328$

$329$

$330$

$331$

$332$

答案：D

知识点：cube 容斥原理 counting

难度评级：1510

小提示：

从一个观察点至多能看到立方体的三个面

At most three faces of a cube are visible from one point

大提示：

对三个两两相邻的 $11\times11$ 面应用容斥原理

Use inclusion-exclusion on three mutually adjacent $11\times11$ faces

解答：

从适当的观察点可以看到三个两两相邻的面。它们共包含 $\begin{aligned} &3(11^2)-3(11)+1\\ &=363-33+1\\ &=331 \end{aligned}$ 个不同的单位立方体：减去三条重复计算的公共棱，再补回顶角处的单位立方体。

所以正确答案是 D。

From a suitable point one can see three mutually adjacent faces. They contain $\begin{aligned} &3(11^2)-3(11)+1\\ &=363-33+1\\ &=331 \end{aligned}$ distinct unit cubes: subtract the three shared edges and restore the corner cube.

Thus the correct answer is D.

1990 AMC 12 第 10 题

答案：D

难度评级：1510

小提示：

大提示：

解答：

其他年份的第 10 题