Dividing a Square Cake into Five Equal Pieces

Date: 07/28/2001 at 10:48:48
From: Pam English
Subject: Dividing regions of a square

How can you divide a square-topped cake that is a rectangular solid 
and is frosted on all faces into five pieces so that everyone receives 
the same amount of cake and icing? All cuts must be perpendicular to 
the surface.  

I can not figure this out. All I can do is divide the cake four ways 
and eight ways. Please help me or tell me a way that I can try to 
solve this problem.

Date: 07/28/2001 at 22:20:31
From: Doctor Peterson
Subject: Re: Dividing regions of a square

Hi, Pam.

This problem is easier to solve than you would think. I suggest you 
start by thinking about wedge-shape pieces like this:

            s
    +---------------+
    |               |
    |               |
    |               |
    |       +       |
    |      / \      |
    |     /   \     |
    |    /     \    |
    +---+-------+---+
            a

Then you can extend it to wedges that wrap around a corner like this:

            s
    +---------------+
    |               |
    |               |
    |               |
    |       +       |
    |      /    \   |
    |     /         +
    |    /          |b
    +---+-----------+
              a

You can find this area by drawing the line from the center to the 
corner to divide it into two triangles, and finding the area of each 
triangle.

Once you've done this, I think you will have a good idea what to do.

For a more detailed answer in our Dr. Math archives, see:

   Cutting a Square into Five Equal Pieces
   http://mathforum.org/dr.math/problems/shinichi.7.12.99.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/