2008 AMC 12A Problem 11

Problem 11 of 25IntermediateAlgebra Geometry

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the

13

visible numbers have the greatest possible sum. What is that sum?

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Solution

The six faces of each cube sum to

1 + 2 + 4 + 8 + 16 + 32 = 63.

From the pattern, the pairs of opposite faces are

1

32,

2

16,

and

4

8.

Each of the two lower cubes hides a pair of opposite faces (top and bottom); hiding the pair

4 + 8 = 12

is best. The top cube hides only its bottom face, so hide the

1.

The greatest sum is

\begin{aligned} &3 \cdot 63 - 2 \cdot 12 - 1 \\ &= 189 - 24 - 1 \\ &= 164. \end{aligned}

Thus, C is the correct answer.