Cutting Cake with Geometry

Date: 2/4/96 at 4:37:18
From: Mittledorf Joshua J
Subject: Geometry puzzle

Here's a cute geometry puzzle that came to me from a very public 
source, which I'm too embarrassed to credit:

A cake is square when viewed from the top.  Height is unspecified.  
It is iced on top and the four vertical sides.  How can the cake be 
divided in 5 pieces such that each piece has the same amount of cake and 
the same amount of icing?  How can you minimize the number of cuts you 
have to make in the cake and still meet this target?  Alternatively, how 
can you minimize the total length of the cuts you make in the cake? 

- Josh Mitteldorf

Date: 2/12/96 at 12:46:37
From: Doctor Ken
Subject: Re: Geometry puzzle

Hello -

You listen to Car Talk too?

When I heard this puzzle on National Public Radio, I thought of this 
solution: the cake has a certain perimeter whn viewed from the top.  Let 
the perimeter be called "P."  Then find five points around the perimeter 
such that they divide the perimeter into five equal parts (each with the 
same length, in other words).  Now if you cut the cake into five 
pieces, and make sure that the corners of the pieces are these five 
points and the tops of the five pieces have the same surface area, then 
you'll have done it!

The other parts of the puzzle can be gotten by drawing pictures and 
messing around with them: for example, you know it's going to take at 
_least_ three cuts, because two cuts can only get you four pieces.

-Doctor Ken,  The Math Forum