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imc

prealgebra intermediate

Problem

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 . What is the value of the cube?
(A)
(B)
(C)
(D)
Solution
Each of the vertices is counted times because each vertex is shared by three different edges. Each of the edges is counted times because each edge is shared by two different faces. Since the sum of the integers assigned to all vertices is , the final answer is
Final answer
D