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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:
The next slice will be from the left hand side (shown in a different colour again):
So the next cut is from the top. Hard to cut this so I would have put it on its side!
I do three more cuts to get to the $3$ by $3$ by $3$ and these leave the block like this:
If we keep all the slices and the last little cube, we will have pieces that look like (seen from above):
Investigate sharing these thirteen pieces out so that everyone gets an equal share.
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.