Problem:
A cube is formed by gluing together standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is .) The smallest possible sum of all the numbers showing on the surface of the cube is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are six dice that have a single face on the surface, and these dice can be oriented so that the face with the is showing. They will contribute to the sum. There are twelve dice that have just two faces on the surface because they are along an edge but not at a vertex of the large cube. These dice can be oriented so that the and are showing, and they will contribute to the sum. There are eight dice that have three faces on the surface because they are at the vertices of the large cube, and these dice can be oriented so that the , and are showing. They will contribute to the sum. Consequently, the minimum sum of all the numbers showing on the large cube is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions