2005 AMC 10A Problem 11

Question

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

Each of the

n^3

unit cubes has

6

faces, so there are

6n^3

faces in all

Only the original surface is red, contributing

6n^2

red faces; set

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

The unit cubes have

6n^3

faces total, of which the original surface accounts for

6n^2

red faces. Then

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

so

n = 4.

Thus, the correct answer is B.

Problem 11 in other years