Distinct numberings of a cube up to rotation

Question

You erase the numbers from a standard six-faced die and write $1$ through $6$ on its faces again, one per face. Treating two labellings as the same if one can be rotated into the other, how many different dice can you make?

Short answer

$30$ . There are $6!=720$ labellings and the cube has $24$ rotations, each producing a different labelling, so $720/24=30$ . Equivalently: put $1$ on top, choose its opposite ( $5$ ways), and arrange the other four cyclically ( $3!$ ways).

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Sample Spaces, Events, and the Probability Axioms

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