Food for Thought

Here's a thought - take a slice of cheese. With n number of cuts, what is the maximum number of pieces  you could cut your cheese into? Observe...

Notice how as we add each cut, starting at 0 cuts with 1 slice, we get 1, 2, 4, 7, 11... 

A lovely pattern can be seen in the maximum number of pieces you get where you take the previous number and add 1, then 2, then 3, then 4... perhaps more familiarly as the triangular numbers + 1 or n(n+1)/2 + 1

Continuing with that thought, what would happen if you extended this problem to say, a block of cheese? (going from 2D to 3D). Given a certain number of planar cuts, what is the maximum number of pieces you can create? With 2D cheese, when adding a cut, to maximize the number of pieces you get, your line has to cut through each of the existing cuts/lines. The same is true in the case of 3D cheese, in that each new plane must cut all of the existing planes. The following pattern emerges: 

Notice how with 2D cheese, each time you were adding 1, then 2, 3, etc. but then in 3D cheese, the numbers your adding each time are the numbers of pieces per cut from the 2D cheese problem. With the previous formula, you can derive the following pattern in 3D: (n^3+5n+6)/6

Additional notes: 

• This is actually called the lazy caterer's sequence in 2D and cake numbers in 3D 
• It's pretty cool how for n dimensions, the first n slices will be consecutive powers of 2, and the next one will be 2^(n+1)-1
• For any given number of slices, can you cut the 2D or 3D cheese into equally sized pieces? If so, how, and if not, why not? (I actually have no idea)