2021 Fall AMC 12B Problem 20

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

By Burnside’s lemma, the count is the average number of

4

-blue colorings fixed by each of the

24

rotations acting on the

8

cubies. The identity fixes

\binom{8}{4} = 70.

The

6

face quarter-turns fix

2

each

(12).

The

3

face half-turns fix

6

each

(18).

The

8

vertex rotations fix

4

each

(32).

The

6

edge half-turns fix

6

each

(36).

The total is

70 + 12 + 18 + 32 + 36 = 168,

and

\dfrac{168}{24} = 7.

Thus, the correct answer is A.