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Sides of a Cube Problem

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Eight small cubes are put together to form one large cube. All six sides of the larger cube are painted, the paint is allowed to dry, and then the cube is taken apart.

a) How many of the small cubes will have paint on just one side? On two sides, On three sides? On no sides?
b) Complete the following table, assuming in turn that the large cube is formed from 27 and 64 cubes instead of eight. See the table attached.
c) Briefly explain the thinking and patterns you see.

Explain.

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The problem is solved in around 270 words with calculations to aid the reasoning.

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a) Let p(k) means the number of small cubes that will be painted k sides. When the big cube is made of eight small cubes, all small cubes are at the corner position and therefore, they will be painted three sides, p(3)=8.

b) For 27 cubes, we have a big cube made of 3x3x3 small cubes. Again, we have eight small cubes ...

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