2021 Fall AMC 10A Problems/Problem 24

Problem

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 $2$ ?

$\textbf{(A) } 8 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 20$

Solution

Since we want the sum of the edges of each face to be $2$ , we need there to be $2$ $1$ s and $2$ $0$ s. Through experimentation, we find that the $1$ s and $0$ s must be adjacent to each other on the faces.

I am still working on the solution - in the meantime PLEASE DO NOT EDIT.

~Arcticturn

2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2021 Fall AMC 10A Problems/Problem 24

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