2021 AMC 12B Fall Problem 20
Attempt Problem 20 of the 2021 AMC 12B Fall 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 2021 AMC 12B Fall solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
20.
A cube is constructed from white unit cubes and blue unit cubes. How many different ways are there to construct the cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Answer: A
Small Hint:
Use Burnside’s lemma over the rotations of the cube acting on the corner positions
Big Hint:
For each rotation type, count the colorings with exactly blue cubes that it fixes
Solution:
By Burnside’s lemma, the count is the average number of -blue colorings fixed by each of the rotations acting on the cubies.
The identity fixes The face quarter-turns fix each The face half-turns fix each The vertex rotations fix each The edge half-turns fix each
The total is and
Thus, the correct answer is A.
Problem 20 in Other Years
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