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The entire exterior of a large wooden cube is painted red

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Re: The entire exterior of a large wooden cube is painted red  [#permalink]

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New post 05 Aug 2019, 19:18
TheGerman wrote:
The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2
B. 6n^2 – 12n + 8
C. 6n^2 – 16n + 24
D. 4n^2
E. 24n – 24


The number of smaller cubes that have no faces painted is (n - 2)^3. Therefore, the number of smaller cubes that have at least one face painted is:

n^3 - (n - 2)^3 = n^3 - (n^3 - 6n^2 + 12n - 8) = 6n^2 - 12n + 8

Alternate Solution:

Let n = 3. Then we know we have 3^3 = 27 smaller cubes, and 26 of them (all except the innermost cube) will have at least one face that is painted red.

If we plug in 3 for n in each answer choice, we see that choice B is the only one that gives 26 as the answer: 6 * 9 - 12 * 3 + 8 = 54 - 36 + 8 = 26.

Answer: B
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Re: The entire exterior of a large wooden cube is painted red   [#permalink] 05 Aug 2019, 19:18

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