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:

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

Thus, the answer is A .

2021 AMC 10B Fall Problem 24

Solution:

Problem 24 in Other Years