2002 AMC 10A Problem 18

Question

A

3\times 3\times 3

cube is formed by gluing together

27

standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is

7.

) The smallest possible sum of all the numbers showing on the surface of the

3\times 3\times 3

cube is

60

72

84

90

96

Classify the

27

dice as corner, edge, face-center, and interior dice

Minimize each die’s showing faces: corners show

1+2+3,

edges show

1+2,

face-centers show

1

The

8

corner dice show

3

faces each, minimized at

1+2+3=6,

contributing

8\cdot 6=48.

The

12

edge dice show

2

faces, minimized at

1+2=3,

contributing

12\cdot 3=36.

The

6

face-center dice show

1

face, minimized at

1,

contributing

6,

and the hidden interior die contributes

0.

The total is

48+36+6=90.

Thus, the correct answer is D.

Problem 18 in other years