Derivation of Patition Formula

Hi ,

I was working on a problem ( as a hobby ) .

It wanted the maximum number of pieces a cake could be sliced into with ' n ' slices .

The formula is Max( P) = n(n+1)/2 + 1

where ' P ' is the number of pieces , ' n ' the number of slices .

Can anyone suggest a simple way if deriving it ?

The nth cut intersects with at most n-1 other cuts (all the previous ones), giving at most (n-1+1) = n more slices.

This gives Max = 1 + sum{1 <= i <= n} i = n(n+1)/2 + 1.

That's still not all of it. You have to show (or assume) that no two intersection points coincide.

Forget the cake, slice a doughnut!

How many components can be formed with n planar cuts through a torus? (You may treat the locations of the planes _and_ both radii as variables.)

I see no one went for this puzzle. With n=1 you can have up to two components, of course. By the time you move to n=3 it is possible to have 13 components. I don't know if there is a formula for all n; I would be surprised if there were.

A reference: www.math-atlas.org/99/bagel_cut (from the last time this came up...)

: I see no one went for this puzzle. With n=1 you can have up to two : components, of course. By the time you move to n=3 it is possible to have : 13 components. I don't know if there is a formula for all n; I would : be surprised if there were.

: A reference: www.math-atlas.org/99/bagel_cut (from the last time this : came up...)

: dave

According to http://mathworld.wolfram.com/TorusCutting.html the number of components is (1/6)(n^3+3n^2+8n).

Surfing around Mathword, I ran across the Pizza Theorem ...

Q: What's the volume of a pizza of thickness 'a' and radius 'z'?

A: http://mathworld.wolfram.com/PizzaTheorem.html