染色立方体

Painted Cube

专题: Brainteaser / 脑筋急转弯
难度: L2
来源: OpenQuant

题目详情

你有一个 $10$ × $10$ × $10$ 立方体，由各个未上漆的单位立方体制成。如果将整个立方体浸入油漆中，有多少个立方体没有接触任何油漆？

You have a $10$ by $10$ by $10$ cube made out of individual unpainted unit cubes. If you dip the entire cube in paint, how many individual cubes did not touch any paint?

解析

如果移除立方体的外部油漆覆盖外壳，你将留下一个未接触任何油漆的 $8$ × $8$ × $8$ 立方体。该立方体由 $8^3$ 单个单元立方体组成，因此 $\boxed{512}$ 立方体没有接触任何油漆。

Original Explanation

If you remove the outer paint covered shell of the cube, you are left with an $8$ by $8$ by $8$ cube that has not touched any paint. This cube is made up of $8^3$ individual unit cubes so $\boxed{512}$ cubes have not touched any paint.