2023 AMC 10A Problem 6

Question

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is

21.

What is the value of the cube?

42

63

84

126

252

Answer: D

Concepts:double counting cube geometry

Difficulty rating: 1270

Small hint:

Each edge belongs to

2

faces, so the total face value is

2

times the total edge value

Big hint:

Each vertex belongs to

3

edges, so the total edge value is

3

times the vertex total

Solution:

Count by incidences. Each edge lies on

2

faces, so the six face values together are

2

times the total of all edge values. Each vertex lies on

3