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Painting high dimension cubes (Posted on 2015-03-05) Difficulty: 3 of 5
If you paint the six sides of a 3x3x3 cube and then slice it into 27 unit cubes, how many will have paint on 0, 1, 2, 3 of their square sides?

Generalize to any dimension:
If you paint the sides of a n dimensional cube with sides length 3 and slice it into 3n unit cubes, how many will have paint on 0, 1, 2, ..., n of their (n-1) dimensional sides?

No Solution Yet Submitted by Jer    
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Some Thoughts Well this seems pretty straightforward. | Comment 1 of 4
0 sides = interior cubes = (n-2)^3
1 side = 6 sides * (n-2)^2 per side = 6(n-2)^2
2 sides = 12 edges *(n-2) per edge = 12(n-2) 
3 sides = corners = 8

Checking, we see that this adds correctly:

n^3 -6n^2 + 12n + 8
        6n^2 - 24n  +24
                   12n  - 24
                             8
----------------------------
n^3

/***********************************/
oops.  Misread the problem.  I obviously answered for a 3 dimensional cube of side n.

Edited on March 5, 2015, 3:03 pm
  Posted by Steve Herman on 2015-03-05 14:56:53

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