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Holes

Age 5 to 11 Challenge Level:

1st pic

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?