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?
Answer Choices:
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Solution:
Each vertex of the cube touches 3 edges, so the sum of the values of the 12 edges is 3 times the sum of the values of the 8 vertices. Each edge touches 2 faces, so the sum of the values of the faces is twice the sum of the values of the edges. Thus the value of the cube is .
For example, assigning 21 to one vertex and 0 to the other seven vertices will result in the given values. Specifically, there will be three edges with value 21 and nine edges with value 0 . This means that there will be three faces with value 42 and three faces with value 0 .
The problems on this page are the property of the MAA's American Mathematics Competitions