Challenging Problem from nostressprep.com

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Hi beatthegmat community!

I'm trying this problem out as one of the challenge problems on my blog/website-- nostressprep.com.

Trying to get a good sense of what level GMAT problem this question would be at. As a GMAT
tutor, I'm guessing somewhere in the upper 700's, but as I am the creator of the problem
I may be a bit biased. Anyways, here it is-- let's see if anyone can get it!


A cube is to be constructed from 8 smaller, equally sized cubes. How many possible ways can the larger cube be arranged taking into account the position and orientation of each of the smaller cubes?


(A) (8^2) (4^1) (2^1)
(B) 6^8
(C) 4! X6^8 X 2^7
(D) 4! (2X3) ^8
(E) 35 (2^15 X 3^10)
Source: — Problem Solving |

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by Maciek » Sun Sep 05, 2010 7:36 am
Hi Chris!

This is great question.

IMO E

We need to divide this problem into two problems.
How many possible ways can the larger cube be arranged taking into account the position of each of the smaller cubes?
We have 8 smaller cubes, so it is 8!

How many possible ways can the larger cube be arranged taking into account the orientation of each of the smaller cubes?
We have 8 smaller cubes and 6 orientations, so it is 6^8

The answer is:
8!*6^8 = 8*7*6*5*4*3*2*1*(2*3)^8 = (2^3)*7*(2*3)*5*(2^2)*3*2*(2^8)*(3^8) = 2^15*3^10*7*5 = 35*(2^15*3^10)

It is answer E

Hope it helps!
Best,
Maciek
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by [email protected] » Sun Sep 05, 2010 7:52 am
Great Maciek!

The answer is E.

The last part can be a bit tricky-- breaking up 8!. Seeing that none of
the other answers have 8! factorial (most only have 4!), you can arrive at
(E) by process of elimination.

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by Maciek » Sun Sep 05, 2010 8:09 am
Chris, it was really tricky:)
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by Ian Stewart » Sun Sep 05, 2010 9:08 am
[email protected] wrote:Great Maciek!

The answer is E.

The last part can be a bit tricky-- breaking up 8!. Seeing that none of
the other answers have 8! factorial (most only have 4!), you can arrive at
(E) by process of elimination.
There are more than six ways to orient a cube: once you choose which face will be the top face (6 choices) you can rotate the cube in 4 ways, so there are 24 different orientations of each cube. So the answer here (unless my interpretation of the question is different from what's intended) should be 24^8 * 8!.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by [email protected] » Wed Sep 08, 2010 9:56 am
I realize that I did ask for the orientation of the cube, so 24^8--as you'd pointed out Ian--is correct. What I was asking for was the number of ways a cube can be positioned so that a face is in a distinct position regardless of the position of the other faces. Since I didn't explicitly state this, and I did ask for orientation, Ian's response is correct.

At least, I don't think there is any ambiguity in the original question.

It just looks like the question was even more diabolical then I'd intended :).

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by [email protected] » Wed Sep 08, 2010 9:57 am
Also, posting another question today. Hopefully, this one is what I'd intended.