2000 AMC 12 Problem 25
Attempt Problem 25 of the 2000 AMC 12 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 2000 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
25.
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
Answer: E
Small hint:
There are ways to place the eight distinct colors before accounting for rotations
Big hint:
The rotation group of the octahedron has elements, and distinct colors give each arrangement a trivial stabilizer
Solution:
There are ways to assign the eight distinct colors to the eight faces. Two assignments give the same octahedron exactly when one is a rotation of the other.
The rotation group of a regular octahedron has elements. Because all eight colors are different, no nontrivial rotation fixes a coloring, so each distinguishable octahedron corresponds to exactly assignments.
Therefore the number of distinguishable octahedrons is
Thus, the correct answer is E.
Problem 25 in other years
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