SOLUTION: A wooden cube whose edges are 4 inches is painted green.The cube is then cut into 64 one-inch cubes. How many small cubes have exactly 1 green face?

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Question 331949: A wooden cube whose edges are 4 inches is painted green.The cube is then cut into
64 one-inch cubes. How many small cubes have exactly 1 green face?
Answer by galactus(183) About Me  (Show Source):
You can put this solution on YOUR website!
The number of one side paintings on an nxnxn cube is
6%28n-2%29%5E2
In this case, we are told that n=4. So we have:
6%284-2%29%5E2=24
There are 24 little cubies with only one side painted.
The number with two sides painted is 12%28n-2%29
12%284-2%29=24
The number with three sides painted is always 8. They are on the corners and that is true regardless of the cube size.
Those remaining are the ones in the center with no paint. That is
%28n-2%29%5E3
%284-2%29%5E3=8
So, we have 24+24+8+8=64...as it should be.
Add up the formulas:
8%2B%28n-2%29%5E3%2B6%28n-2%29%5E2%2B12%28n-2%29
A pattern. A binomial. Add these up and it reduces all the way down to
n%5E3
Just as we expected. 4^3=64