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骰子标号

Dice Labeling

专题
Probability / 概率
难度
L3

题目详情

如果把一个六面骰上的数字全部擦掉，再重新给六个面标上数字，那么一共有多少种不同的标号方式？通过旋转骰子可以互相得到的标法视为相同。

How many distinct ways can you label a 6 sided dice if you wipe off all the numbers? Arrangements that can be formed by rotating the die around are not considered distinct.

解析

先看立方体的旋转。一个骰子共有 24 种不同的空间旋转方式：固定某一面朝上时，这一面还可以有 4 种转法，而可选的“朝上面”共有 6 个，所以总共是 4×6=244\times 6=24。

如果直接把 1 到 6 标到 6 个面上，一共有 6!6! 种标法；但其中每一种实际上都把同一个骰子的 24 种旋转姿态重复算进去了。因此不同的标号方式数为

6!24=30\frac{6!}{24}=\boxed{30}

Original Explanation

First, notice that there are 2424 ways of turning the die. This is because if you consider one particular face of the die, there are exactly 44 ways of turning it so that this face is always in the same place. Then do the same for the other faces, and get 4×6=244 \times 6=24

6!6! is the number of ways to number the die but this also counts the 2424 possible turns of the die. So the answer is 6!24=30\frac{6!}{24}=\boxed{30}