# Number of painted stickers on a cube



## Werner (Aug 30, 2009)

Imagine a a 3x3x3 cube ( not a rubiks cube ! ) you split this cube into 27 1x1x1x cubies. Now you paint the outside of the cube, how many stickers are painted on each side ? 

Now it does not take a genius to figure out that:
1 cube is not painted
8 cubes are painted on 3 sides
12 cubes are painted on 2 sides
6 cubes are painted on one side 

Now solve the polynomial (2x + 1)^3 

Can you based on this figure out how the distrubution of paint is on a 4x4x4 ? 

Bonus question: Same question above but form a general law for nxnxn cubes. 

Atleast I found this interesting, and it is related to puzzle theory...


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## Lucas Garron (Aug 30, 2009)

Werner said:


> Now solve the polynomial (2x + 1)^3


You mean "expand," not "solve."

I actually find this interesting, though.

(2x+n-2)^3

Comes from:
8 * x^3 +
12 * (n - 2) * x^2 +
6 (n - 2)^2 * x+
(n - 2)^3

(Which factors.)


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## Lucas Garron (Aug 30, 2009)

Hmm, I get:

(-2 + n + 2 x) (17 + 5 n^2 + 20 n (-1 + x) - 40 x + 20 x^2) / 2

for minx, assuming there are smaller minxes nested inside each other.

(n=3 is mega, n=5 is giga, etc.)


TEGTaylor and mr.onehanded: Stop it.
(If I had mod powers, I would seriously delete your posts.)

EDIT: David Woner deleted the posts (thanks). They were squabbling about "painted" stickers.


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## Werner (Aug 31, 2009)

4x4x4 is (2x + 2)^3

Supercube would be interesting, sorry about my confusing english. Its not my mother tounge.

(2x + 0)^3 = 8x^3 

this is for a 2x2x2


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## Lucas Garron (Aug 31, 2009)

Werner said:


> 4x4x4 is (2x + 2)^3
> 
> Supercube would be interesting, sorry about my confusing english. Its not my mother tounge.
> 
> ...



Did you see this line? I should probably have specified that it was the nxnxn formula:


Lucas Garron said:


> (2x+n-2)^3



What do you mean with supercube? The normal definition is a cube with special stickers, but those stickers are in exactly the same places.


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