Holes

'Holes' printed from https://nrich.maths.org/

Here we have three solid cubes and three cubes that have holes.

They're just the first three in a series that could go on and on.

I was wondering about the number of cubes used in each ...

Then I thought about the difference between those numbers.

So, for example, I found that the first cube, $3$ by $3$ by $3$, used $27$ cubes.

The same cube with holes used $20$ cubes, a difference of $7$.

You can call these types of cubes what you like but I called them "Solid" and "Frame".

How about exploring the numbers for the next few Solid and Frame cubes?

Do you notice any patterns?

Can you explain any of the patterns?