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Target Test Prep Representative V
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Re: The entire exterior of a large wooden cube is painted red  [#permalink]

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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.

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

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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

problem never specifies a real number for any of the steps, so select a smart number for n, say n = 3. In this case, the original cube is painted and then cut into a 3 × 3 × 3 assemblage of 27 smaller cubes. Imagine the top face of the original (large) cube (if you know what a Rubik’s Cube is, picture one!). Every cube on that face has been painted on at least one side. The same is true for the bottom face. Now think about the middle “slice” of the cube. This “slice” contains a total of 9 cubes, but only the one in the very middle has not been painted on any of its sides.

Therefore, only one of those 27 cubes—the one in the middle of the structure—remains unpainted. When n = 3, the answer is 26.

Plug n = 3 into the answers and look for 26:

(A) 6(32) = 6(9) = 54
(B) 6(32) – 12(3) + 8 = 54 – 36 + 8 = 26
(C) 6(32) – 16(3) + 24 = 54 – 48 + 24 = 30
(D) 4(32) = 4(9) = 36
(E) 24(3) – 24 = 72 – 24 = 48
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Re: The entire exterior of a large wooden cube is painted red  [#permalink]

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Bunuel wrote:
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

Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:
Attachment:
Red Cube.png

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

How is 6n^2 =54 if n=3?

Math Expert V
Joined: 02 Sep 2009
Posts: 65062
Re: The entire exterior of a large wooden cube is painted red  [#permalink]

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1
mozerng wrote:
Bunuel wrote:
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

Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:
Attachment:
Red Cube.png

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

How is 6n^2 =54 if n=3?

6n^2 = 6*3^2 = 6*9 = 54.
_________________ Re: The entire exterior of a large wooden cube is painted red   [#permalink] 07 May 2020, 02:59

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