The Big Cheese

I met up with some friends yesterday for lunch. On the table was a good big block of cheese. It looked rather like a cube. As the meal went on we started cutting off slices, but these got smaller and smaller! It got me thinking ...

What if the cheese cube was $5$ by $5$ by $5$ and each slice was always $1$ thick?

It wouldn't be fair on everyone else's lunch if I cut up the real cheese so I made a model out of multilink cubes:

You can see that it's a $5$ by $5$ by $5$ because of the individual cubes, so the slices will have to be $1$ cube thick.

So let's take a slice off the right hand side, I've coloured it in so you can see which bit I'm talking about:

This now gets cut off and we have:

The next slice will be from the left hand side (shown in a different colour again):

Well off it goes with the knife and we are left with:

I'm setting myself the task of cutting so that I am left with as close to another cube shape as possible each time.

So the next cut is from the top. Hard to cut this so I would have put it on its side!

I'll remove that and I'm left with the $4$ by $4$ by $4$ cube:

I do three more cuts to get to the $3$ by $3$ by $3$ and these leave the block like this:

I'm sure you've got the idea now so I don't need to talk as much about what I did:

and then onto:

That leaves you with this smallest cube of $1$ by $1$ by $1$.

If we keep all the slices and the last little cube, we will have pieces that look like (from above):

Now we have thirteen objects to explore. You could look at the area as seen from above, total surface area and volume of the pieces.

You could investigate trying to share these thirteen pieces out so that everyone gets an equal share.

You could try using the thirteen cuboids to re-construct the $5$ by $5$ by $5$ cube and explore different ways of doing that!

What about ...?

I guess that once you've explored the pattern of numbers you'll be able to extend it as if you had started with a $10$ by $10$ by $10$ cube of cheese.