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.)

$7$

$8$

$9$

$10$

$11$

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.

2021 AMC 12B Fall Problem 20

Solution:

Problem 20 in Other Years