2005 AMC 12A Problem 10

Attempt Problem 10 of the 2005 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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10.

A wooden cube

n

units on a side is painted red on all six faces and then cut into

n^3

unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is

n?

3

4

5

6

7

Count the total number of small-cube faces and the number that are red

There are

6n^3

faces in all and

6n^2

red ones

The

n^3

unit cubes have

6n^3

faces total. The red faces are exactly the surface of the original cube,

6n^2

of them.

Setting the red fraction to one-fourth,

\dfrac{6n^2}{6n^3} = \dfrac{1}{n} = \dfrac{1}{4},

n = 4.

Thus, the correct answer is B.

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