2005 AMC 10A 第 11 题

Question

一个边长为

n

个单位的木质立方体六个面全部涂红，然后切成

n^3

个单位小立方体。所有小立方体面的总数中，正好有四分之一是红色的。

n

是多少？

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

n^3

个单位立方体每个都有

6

个面，所以总面数为

6n^3

。

Each of the

n^3

unit cubes has

6

faces, so there are

6n^3

faces in all

只有原立方体表面是红色的，贡献

6n^2

个红色面；令

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

。

Only the original surface is red, contributing

6n^2

red faces; set

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

小立方体共有

6n^3

个面，其中来自原立方体外表面的红色面共有

6n^2

个。因此

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

，所以

n = 4

。

所以正确答案是 B。

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.

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