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## 'Holes' printed from http://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?