Partially Painted Cubes

Introduction

The following task is taken from nRich and it is quite challenging!

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.

Either explore the questions above on your own or look at the following section for some prompts.

You may wish to colour in squares on these templates (PRINT ME) to help visualise this.
If only one face is painted of the large cube is painted, how many small cubes are not painted?
If two faces of the large cube are painted, how many small cubes are not painted?
- How many different ways can two faces be painted?
- What constitutes a "different" way of painting the faces?
What if three faces are painted?
- How many different ways can this be done?
- How many small cubes are not painted?
If four faces are painted, how can you use the answer from the "two faces" problem to help work out the solution?
What does the last bullet point suggest regarding painting five and six faces respectively?

Now repeat for a 4x4x4 cube. What are the possible numbers of unpainted cubes depending on whether 1, 2, 3, 4, 5 or all 6 faces are painted?
Now repeat for a 5x5x5 cube and after that for a 6x6x6 cube. What have you noticed?
What if it is a nxnxn cube? What have you noticed?
- Hint: the number of unpainted cubes can always be expressed as the product of three factors. What can you say about these factors?
You should now go back to the original question with Jo and Dan.

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 others that it is impossible to end up with 50 unpainted cubes?
What if we extend to a 2x3x4 cuboid? What are the possible numbers of unpainted cubes depending on whether 1, 2, 3, 4, 5 or all 6 faces are painted?
Try other cuboids with different dimensions. Can you generalise a rule for an p by q by r cuboid?
What if we now extend to other regular polyhedra such as tetrahedra, octahedra, etc. What have you noticed?