Question

Each of the $12$ edges of a cube is labeled $0$ or $1.$ Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$

$8$

$10$

$12$

$16$

$20$

Solution:

Label one face $ABCD$ . Each face must contain two $0$ s and two $1$ s, so first split by the pattern on the four edges of $ABCD$ .

Case 1: opposite edges of $ABCD$ have the same label. There are $2$ such patterns on $ABCD$ . For either pattern, once the label of one vertical edge, say $AE$ , is chosen, the face-sum conditions force all remaining labels. This gives $2\cdot2=4$ labelings.

Case 2: opposite edges of $ABCD$ have different labels. There are $4$ such patterns on $ABCD$ . For each, the labels of two adjacent vertical edges may be chosen in $4$ ways, and then the remaining labels are forced by the face-sum conditions. This gives $4\cdot4=16$ labelings.

The total is $4+16=20$ .

Thus, E is the correct answer.

2021 AMC 10A Fall Problem 24

Solution:

Problem 24 in Other Years