2023 AMC 10A Problem 6

We do not know the eight numbers, only their total. So instead of adding things up, follow one vertex's number and count how many times it ends up in the cube's value.

1. Vertex to edges. Each corner of a cube touches $3$ edges. So a vertex's number is added into $3$ edge values.
2. Edges to faces. Each edge of a cube is the border between $2$ faces. So each edge value is added into $2$ face values.
3. Put these together. A vertex's number goes into $3$ edges, and each of those goes into $2$ faces. So the number shows up $3 \times 2 = 6$ times in the sum of the six faces.
4. Every vertex looks the same, so every vertex's number is counted $6$ times. The cube's value is
$$6 \times (\text{sum of all vertex numbers}) = 6 \times 21 = 126.$$

The answer is $\boxed{\textbf{(D)}\ 126}$.

The value is built in layers: vertices make edges, edges make faces, faces make the cube. Each layer just adds up the one before, so the final total is always the same multiple of the vertex total. To find that multiple, count how many times one vertex is used. Watch for this "how many times is each piece counted" idea whenever a total is built out of shared parts.

Since only the total $21$ matters, pick easy numbers. Put $21$ on one corner and $0$ on the other seven.

• The $3$ edges touching that corner each have value $21$. All other edges are $0$.
• The $3$ faces touching that corner each contain two of those edges, so each face has value $21 + 21 = 42$. The other three faces are $0$.
• The cube's value is $3 \times 42 = 126$.