魔方贴纸

Rubik's Cube Stickers

专题: Probability / 概率
难度: L2
来源: OpenQuant

题目详情

一个 $4 \times 4 \times 4$ 的魔方由 $1 \times 1 \times 1$ 的小方块组成。魔方外层贴有表示颜色的贴纸。问至少带有一张贴纸的小方块有多少个？

A $4 \times 4 \times 4$ Rubik's Cube is composed of $1 \times 1 \times 1$ mini-cubes. The Rubik's Cube has stickers on its outer layer to denote the colors of the cube. How many mini-cubes have at least one sticker on them?

解析

总共有 $4^3$ 个小方块。内部完全看不见、因此没有贴纸的是中间的 $2^3$ 个小方块，所以至少有一张贴纸的小方块数量为：

4^3 - 2^3 = 56

Original Explanation

There are a total of $4^3$ mini-cubes that compose the Rubik's Cube. Of these, the core $2^3$ mini-cubes that make up the inside of the cube and that are not showing do not have stickers on it. Thus, the total number of mini-cubes that do have stickers on it is:

4^3-2^3=56