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The entire exterior of a large wooden cube is painted red

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

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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 14 Jul 2013, 09:38, edited 1 time in total.
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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


Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:
Attachment:
Red Cube.png
Red Cube.png [ 79.39 KiB | Viewed 18160 times ]

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.

Answer: B.
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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


Similar questions to practice:
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a-big-cube-is-formed-by-rearranging-the-160-coloured-and-99424.html
a-large-cube-consists-of-125-identical-small-cubes-how-110256.html
64-small-identical-cubes-are-used-to-form-a-large-cube-151009.html

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

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Obviously bunuel's solution is mindblowing and the best approach we need in such PS questions ...But just while brainstorming trying to find the solution to this problem i reached here ...Try to visualise that all the smaller cubes which lie on the exterior face of the larger wooden cube have one or more faces painted red...rest all other cubes which lie beneath the first layer of cube wont have any faces painted red...

Now if we can visualise the situation .....we can see that if we remove the external layers of cube ..we will be left with cubes having none of their faces red coloured...and if we remove this external layers of cube we are basically removing one cube from each side symmetrically ...so we will be left with a cube having dimensions of (n-2).... so basically we will be left with (n-2)^3 cubes ... now if we want to find the number of cubes in the larger cube having one or more faces as red we can deduct (n-2)^3 from n^3...

so number of cubes painted red =n^3 - (n-2)^3
=n^3-(n^3-8-6n^2+12n)= 6n^2-12n+8.......B

this solution is fairly easy too just need little bit of visualization........
Bunuel please correct me if i am wrong somewhere ...

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

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New post 11 Dec 2013, 05:29
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.

Answer: B.


Hi Bunuel,

As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question.

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

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New post 12 Dec 2013, 03:04
davidfrank 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.

Answer: B.


Hi Bunuel,

As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question.


You could apply the same logic:

Say n=5. So, we would have that the large cube is cut into 5^3=125 smaller cubes. Out of them (5-2)^3=27 little cubes won't be painted red at all and the remaining 125-27=98 will have at least one red face. Now, plug n=5 and see which one of the options will yield 98.

But you can use n directly:
Total = n^3
Not painted: (n-2)^3

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

Hope it helps.
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Number of cube inside = (n-2)^3

(n-2)^3 cubes have no colored faces.

Remaining cubes will have at least one face colored red.

Remaining cubes = n^3 -(n-2)^3

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

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

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Just for purpose of simplification, this formula can be used

\(a^3 - b^3 = (a-b) (a^2 + ab + b^2)\)

\(= (n - n + 2) (n^2 + n(n-2) + (n-2)^2)\)

\(= 2 (n^2 + n^2 - 2n + n^2 - 4n + 4)\)

\(= 2 (3n^2 - 6n + 4)\)

\(= 6n^2 - 12n + 8\)
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New post 14 Sep 2014, 07:57
Hi Guys,
When I try to substitute n = 4, it seems that both B and C works. Please help!

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

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New post 14 Sep 2014, 16:00
maggie27 wrote:
Hi Guys,
When I try to substitute n = 4, it seems that both B and C works. Please help!


For plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.
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Re: The entire exterior of a large wooden cube is painted red [#permalink]

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New post 04 Mar 2015, 08:51
A nice example of plug in method usage.
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Re: The entire exterior of a large wooden cube is painted red [#permalink]

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New post 05 Mar 2015, 08:54
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.

Am I missing something here? 54 - 36 + 8 = 10 (not 26)

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New post 05 Mar 2015, 09:05
sushi574 wrote:
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.

Am I missing something here? 54 - 36 + 8 = 10 (not 26)


54 - 36 + 8 = 26, not 10.
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Re: The entire exterior of a large wooden cube is painted red [#permalink]

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New post 09 Aug 2015, 02:48
kundankshrivastava wrote:
Number of cube inside = (n-2)^3

(n-2)^3 cubes have no colored faces.

Remaining cubes will have at least one face colored red.

Remaining cubes = n^3 -(n-2)^3

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

Answer is B


Is "Number of cube inside = (n-2)^3

(n-2)^3 cubes have no colored faces."

a formula? What's the logic/reasoning behind it? How does one derive this during the exam!

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

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New post 26 Nov 2015, 14:25
The (N-2) approach is right. However 2 is again, an assumption. If I take a side x, the the volume of inner cube becomes \((N-2x)^3\) .

Here's how:

Assuming volume of larger cube =\(N^3\)
Volume of smaller cubes formed = \(N^3/n^3\) or\((N/n)^3\)

Thus side of smaller cubes =\(N/n\)= \(x\)(say)
Therefore, volume of smaller cube =\((N-2x)^3\)

Notice that all the squares formed in this cube won't be touched by the paint.

Therefore, total volume of cubes that will have a face painted = \(N^3-(N-2x)^3\)
And no. of cubes with this volume can be found by dividing the above equation by \(x^3\).

If we start- \((N^3-(N-2x)^3)/x^3\)

= \((8x^3 + blah blah..)/x^3\)
= 8 + something

At this point, we can notice that only one option consists 8 as a constant. Otherwise, if you choose solve it, you get the same result.

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

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New post 01 Oct 2016, 23:16
Im confused about this topic, could someone please help me explain the logic behind, why there's only 1 cube in the middle that's unpainted?

If we look at a rubik's cube ... theres 6 faces .... 9 "smaller cubes" on the face of the cube = 54 smaller cubes

Or on another logic;

3^3 = 27 smaller cubes ----> Rubik's cube has 6 faces , therefore 27/6 = 24 (24 "smaller cubes" on the face of the rubik's cube) and then 3 remaining on the "inside".

Please help me get this logic right in simple language as possible! Bunuel or anybody else.

Thank you!

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

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New post 15 May 2017, 07:20
kundankshrivastava wrote:
Number of cube inside = (n-2)^3

(n-2)^3 cubes have no colored faces.

Remaining cubes will have at least one face colored red.

Remaining cubes = n^3 -(n-2)^3

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

Answer is B


how did you get the formula for: Number of cube inside = \((n-2)^{3}\)?
- something you need to memorize?
- i can't even visualize it...a cube has 6 faces (top/bottom, +4 around). if we assume a particular cube has 3 tiny cubes per row & column, we can conclude there will be an "innermost" tiny cube that need not be painted red (it would be buried inside the cube). how do we know that in order to find this innermost cube, the formula is: \((n-2)^{3}\)

more in-depth analysis would be much appreciated

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

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New post 22 May 2017, 19:52
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


For this problem, pick the smallest n that will satisfy the problem. In this case, n =3 satisfies the criterion. You can then draw a cube with 27 individual pieces. Only the middle cube will be unpainted. So you want to then backsolve from the answer choices until you find 26.

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

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New post 07 Dec 2017, 14:53
Hi All,

This question can be solved by TESTing VALUES. Let's TEST N = 3 (if you think about a standard Rubik's cube, then that might help you to visualize what the cube would look like).

So now every "outside" face of the Rubik's cube has been painted. There's only 1 smaller cube of the 27 smaller cubes that does not have paint on it (the one that's in the exact middle). Thus, 26 is the answer to the question when we TEST N=3. There's only one answer that matches....

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