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## 'Partially Painted Cube' printed from http://nrich.maths.org/

Jo made a cube from some smaller cubes. She painted some of the
faces of the large cube, and then took it apart again.

She counted her cubes and noticed that 45 cubes had no paint on
them at all.

Can you work out how many small cubes Jo
used to make her large cube, and which faces she
painted?

Dan made a cube the same size as Jo's large cube, and also
painted some of the faces.

How many unpainted cubes might Dan have
ended up with?

Now explore the number of unpainted cubes for some other sizes of
cube. Here are some questions you might like to consider:

- If the number of small cubes along each edge is $n$, can you
find expressions for the number of unpainted cubes when you paint
1, 2, 3, 4... faces?
- The number of unpainted cubes can always be expressed as the
product of three factors. What can you say about these
factors?
- There is only one way to end up with 45 unpainted cubes. Are
there any numbers of cubes you could end up with in more than one
way?
- How can you convince yourself that it is impossible to end up
with 50 unpainted cubes?