2021 AMC 12B Fall Problem 20

Question

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.)

7

8

9

10

11

Use Burnside’s lemma over the

24

rotations of the cube acting on the

8

corner positions

For each rotation type, count the colorings with exactly

4

blue cubes that it fixes

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.

Problem 20 in other years