1982 AMC 12 Problem 16

Question

In the adjoining figure, a wooden cube has edges of length

3

meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is

54

72

76

84

86

Answer: B

Concepts:surface area cube geometry 3D geometry

Difficulty rating: 1860

Small hint:

Start with the outer faces after removing their central squares

Big hint:

For each of the three tunnels, count the four interior walls not removed by the crossing tunnels

Solution:

The six outer faces contribute

6(3^2-1^2)=48.

Each of the three length-

3

square tunnels has four inner walls, but the central unit segment of every wall is removed by a perpendicular tunnel, leaving area

2

per wall. Thus the inside contributes

3\cdot4\cdot2=24.

The total is

48+24=72.

Therefore, the correct answer is B.