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# A cube is made up of equal smaller cubes. Two of the sides

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A cube is made up of equal smaller cubes. Two of the sides [#permalink]  19 Mar 2013, 22:17
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A cube is made up of equal smaller cubes. Two of the sides of the larger cube are called A and B. What is the total number of smaller cubes?

(1) When n smaller cubes are painted on A, n is 1/9 of the total number of smaller cubes.
(2) When m smaller cubes are painted on B, m is 1/3 of the total number of smaller cubes
[Reveal] Spoiler: OA

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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  21 Mar 2013, 01:13
This question does not make any sense. Could you please let us know the source of this question.
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  21 Mar 2013, 05:09
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SravnaTestPrep wrote:
A cube is made up of equal smaller cubes. Two of the sides of the larger cube are called A and B. What is the total number of smaller cubes?

(1) When n smaller cubes are painted on A , n is 1/9 of the total number of smaller cubes.
(2) When m smaller cubes are painted on B, m is 1/3 of the total number of smaller cubes

Fact 1: When a cube is made of equal smaller cubes, the number of smaller cubes is $$r^3$$ where r is natural number.

Fact 2: Out of the six sides of the bigger cube, we are considering two sides and have named them A and B resp.

Statement 1: when on side A, n smaller cubes are painted we have that value of n equal to 1/9 of the total number of smaller cubes. So n is divisible by 9. Or the number of smaller cubes is a multiple of 9. Statement is not sufficient to answer the question because for example both 27 smaller cubes and 729 smaller cubes satisfy the conditions that the number of smaller cubes is a multiple of 9 and satisfies fact 1.

Statement 2: In this case the number of smaller cubes is a multiple of 3. The number of smaller cubes cannot be greater than $$3^3$$ based on this statement alone.

Assume first the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 27. In this case each side has 9 smaller cubes exposed. If the number of smaller cubes painted, which is m, is 9.we have m =1/3 of the number of smaller cubes.

Assume next the the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 216. In this case each side has 36 smaller cubes exposed. So you could paint a maximum of 36 smaller cubes. Thus maximum value of m is 36/216 = 1/6 which cannot be equal to 1/3.

For higher multiples of 3, the value of m would keep on decreasing and would never satisfy the conditions mentioned.

Thus statement 2 alone is sufficient to answer this question.

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Last edited by SravnaTestPrep on 25 Sep 2013, 15:56, edited 1 time in total.
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  25 Sep 2013, 10:04
Just to double check, isn't the number of smaller cubes within a larger cube $$r^3$$not $$r^r$$?
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  25 Sep 2013, 15:55
nphilli1 wrote:
Just to double check, isn't the number of smaller cubes within a larger cube $$r^3$$not $$r^r$$?

You are right. Made the correction.
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  30 May 2014, 04:45
SravnaTestPrep wrote:
SravnaTestPrep wrote:
A cube is made up of equal smaller cubes. Two of the sides of the larger cube are called A and B. What is the total number of smaller cubes?

(1) When n smaller cubes are painted on A , n is 1/9 of the total number of smaller cubes.
(2) When m smaller cubes are painted on B, m is 1/3 of the total number of smaller cubes

Fact 1: When a cube is made of equal smaller cubes, the number of smaller cubes is $$r^3$$ where r is natural number.

Fact 2: Out of the six sides of the bigger cube, we are considering two sides and have named them A and B resp.

Statement 1: when on side A, n smaller cubes are painted we have that value of n equal to 1/9 of the total number of smaller cubes. So n is divisible by 9. Or the number of smaller cubes is a multiple of 9. Statement is not sufficient to answer the question because for example both 27 smaller cubes and 729 smaller cubes satisfy the conditions that the number of smaller cubes is a multiple of 9 and satisfies fact 1.

Statement 2: In this case the number of smaller cubes is a multiple of 3. The number of smaller cubes cannot be greater than $$3^3$$ based on this statement alone.

Assume first the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 27. In this case each side has 9 smaller cubes exposed. If the number of smaller cubes painted, which is m, is 9.we have m =1/3 of the number of smaller cubes.

Assume next the the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 216. In this case each side has 36 smaller cubes exposed. So you could paint a maximum of 36 smaller cubes. Thus maximum value of m is 36/216 = 1/6 which cannot be equal to 1/3.

For higher multiples of 3, the value of m would keep on decreasing and would never satisfy the conditions mentioned.

Thus statement 2 alone is sufficient to answer this question.

Sorry to point this out but something here is leading to confusion.

First of all if n is the total number of smaller cubes then each surface would have n/6 cubes exposed. This is what I derive from the green highlighted text where it says if 216 is total number of cubes then each surface would have 36 exposed which means we are doing 216/6= 36

By the same logic if 27 is the total number of smaller cubes then each side should have 27/6 number of cubes exposed which means 4.5cubes.
so you can paint a maximum of 4.5 cubes.

(So how come you say that for 27 total number of smaller cubes each side would have 9 cubes exposed?shouldn't it be 27/6 as you did for 216/6)

4.5/27= 1/6 , this is what we were getting for 216 cubes too!, so we cannot have 1/3 hence it seems that even 27 total smaller cubes is not a possibility.

Please let me know if I have not understood something.
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  04 Jun 2014, 17:37
I am confused.

Quote:
Assume first the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 27. In this case each side has 9 smaller cubes exposed. If the number of smaller cubes painted, which is m, is 9.we have m =1/3 of the number of smaller cubes.

Assume next the the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 216. In this case each side has 36 smaller cubes exposed. So you could paint a maximum of 36 smaller cubes. Thus maximum value of m is 36/216 = 1/6 which cannot be equal to 1/3.

It seems like both Statement 1 and 2 can satisfy the relationship.
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Re: A cube is made up of equal smaller cubes. Two of the sides [#permalink]  05 Jun 2014, 06:40
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pretzel wrote:
I am confused.

Quote:
Assume first the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 27. In this case each side has 9 smaller cubes exposed. If the number of smaller cubes painted, which is m, is 9.we have m =1/3 of the number of smaller cubes.

Assume next the the number of smaller cubes which is a multiple of 3 and satisfies fact 1, is 216. In this case each side has 36 smaller cubes exposed. So you could paint a maximum of 36 smaller cubes. Thus maximum value of m is 36/216 = 1/6 which cannot be equal to 1/3.

It seems like both Statement 1 and 2 can satisfy the relationship.

if i am not mistaken , question may be flawed as I have tried to show above. check this :

Let me know if you see any discrepancies in my explanation.
Thanks
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A cube is made up of equal smaller cubes. Two of the sides [#permalink]  23 May 2015, 13:51
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I solved this problem differently.
We are trying to solve for $$x{^3}$$, the number of smaller cubes that compose the larger cube.

1) On Surface A we paint n cubes where n = $$\frac{1}{9}x{^3}$$. We also know that on any surface there $$x{^2}$$ cubes so the maximum number of cubes we can paint is $$x{^2}$$. This means n = $$\frac{1}{9}x{^3}\le{x{^2}}$$ => $$x\le{9}$$. Since n is an integer we know that $$x{^3}$$must be a multiple of 9. Therefore x=3 OR x=6 OR x = 9. Insufficient.

2) Using the same math as before we get $$x\le{3}$$ with $$x{^3}$$ a multiple of 3 to meet the requirement that n is an integer. The only options we have for x are 1, 2, 3. Only x = 3 gives a multiple of 3. Sufficient.
A cube is made up of equal smaller cubes. Two of the sides   [#permalink] 23 May 2015, 13:51
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