I have a cube and I draw a vertex, a middle of an edge and a diagonal of a face. In how many different ways can I draw them? Two cubes can look similar after a rotation.
I don't know how to start.
2026-04-24 03:47:42.1777002462
In how many different ways can I draw them?
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Ignoring symmetries induced by rotations, you have $8$ choices for the vertex, $12$ choices for the bisected edge, and $6x2=12$ choices for the diagonal, for a total of $8x12x12 = 1152$ ways.
If you want to identify the choices which rotate into each other, I think you can avoid the complexity of Burnside's Lemma since for any choice of (vertex, edge, diagonal), the $24$ cube symmetries give you $24$ different such triples (e.g. the only non-identity rotation that fixes the diagonal moves all $8$ vertices).
So, you only need divide by $24$, giving $1152 / 24 = 48$ ways.