Dice Labels: Rotationally Invariant

How many distinct ways can you label a six-sided die if you remove all the numbers, given that arrangements formed by rotating the die are not considered distinct?

Answer

There are \(6!\) ways to label the die if rotations are considered distinct. Now, to account for rotational symmetry, note that if we fix a particular face (say, label it “1”) on the top, there are 4 ways to rotate the die while keeping that face on top. Since any of the 6 faces can be on top, there are a total of \(6 \times 4 = 24\) rotations. In our context, all these rotations count as one distinct arrangement. Hence, the number of distinct labelings is:

\[ \frac{6!}{24} = \frac{720}{24} = 30. \]
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