2007 AMC 12B Problem 16
Attempt Problem 16 of the 2007 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
16.
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
Answer: A
Small hint:
The tetrahedron has rotational symmetries; apply Burnside’s lemma
Big hint:
Count colorings fixed by the identity, the vertex rotations, and the edge rotations
Solution:
The rotation group of the tetrahedron has elements: the identity, rotations of order about a vertex-face axis, and rotations of order about an edge-midpoint axis.
The identity fixes all colorings. Each vertex rotation fixes one face and cycles the other three, so it fixes colorings; likewise each edge rotation swaps two pairs of faces and fixes
By Burnside’s lemma the number of distinguishable colorings is
Thus, the correct answer is A.
Problem 16 in other years
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