AMC 10 · 2007 · #16
Grade 8 countinggeometry-3dPick an answer.
The natural label for a coloring is its color tally: how many faces are red, white, and blue. Listing all possible tallies is a short systematic list (Tool #2). But a label only counts colorings correctly if it is honest in both directions, and only one direction is easy. Rotations obviously preserve the tally, so different tallies are never confused. The reverse claim — equal tallies force equal appearance — is the one that has to be earned, because a tetrahedron has only 12 rotations while four faces admit 24 rearrangements. So the plan is: build the list, then take an explicit inventory of the twelve rotations (Tool #10) and use a repeated color plus a half-turn (Tool #17) to close the missing half. A final pass over the answer choices (Tool #3) shows exactly which shortcut produces each trap.
What a rotation can change
A rotation never changes the tally of each colour.
Turning the solid shuffles which face sits where, but it never repaints anything, so the color counts cannot move.
Turning the solid shuffles which face sits where, but it never repaints anything, so the colour counts cannot move.
▸ Why?
A rotation moves the solid onto itself without stretching, so it only rearranges faces that already exist.
▸ Why?
Each face is carried to exactly one face, so the tally of each colour is passed along unchanged.
List every color tally
Listing every tally gives 15 possibilities.
Sorting the tallies by shape keeps the list complete with no repeats, and stars and bars checks the same total a different way.
7.SP.C.8Make A Systematic ListWhich rearrangements are rotations
Not every rearrangement of faces is a rotation.
You can list every way to set a tetrahedron down and spin it, and there are only twelve — half of what free relabeling would allow.
8.G.A.1Create A Physical RepresentationA repeated color closes the gap
A repeated colour supplies the swap needed to close the gap.
Two faces of the same color are interchangeable for free, and that free swap converts a forbidden rearrangement into an allowed one.
8.G.A.1Visualize Spatial RelationshipsCount the classes and answer
So the tally count is the answer, 15, choice (E).
Once the tally is proved to be a complete and honest label, counting colorings is just counting tallies.
6.EE.A.1Eliminate PossibilitiesCount how many faces get each color: for a tetrahedron with three colors that tally tells the whole story, because two faces sharing a color can always be swapped by a real turn.
- What a rotation can change
- List every color tally
- Which rearrangements are rotations
- A repeated color closes the gap
- Count the classes and answer
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