2021 Fall AMC 10B Problem 24

A cube is constructed from

4

white unit cubes and

4

blue unit cubes. How many different ways are there to construct the

2 \times 2 \times 2

cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

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Solution

This is the number of ways to choose which

4

of the

8

cube vertices are blue, up to cube rotation. Use Burnside’s lemma on the

24

rotations of the cube. The identity fixes

\binom84=70

colorings. The

6

quarter-turn face rotations each fix

2.

The

3

half-turn face rotations each fix

6.

The

8

rotations about opposite vertices each fix

4.

The

6

half-turn rotations about opposite edges each fix

6.

Thus the number of inequivalent constructions is

\begin{gathered} \frac{70+6\cdot2+3\cdot6+8\cdot4+6\cdot6}{24}\\ =7. \end{gathered}

Thus, the answer is A .