Cutting a Cake into 8 Pieces with 3 Cuts

Date: 09/08/2002 at 23:14:43
From: Matt Seymore
Subject: Cutting a Cake

How do you cut a cake into 8 pieces making only 3 cuts? How do you cut 
a doughnut into 12 pieces with only 3 cuts?

Date: 09/09/2002 at 13:02:14
From: Doctor Peterson
Subject: Re: Cutting a Cake

Hi, Matt.

Let's think about what happens when you make a cut. If the thing we 
are cutting is convex (that is, it has no indentations), then it will 
be cut into exactly two pieces (if it is cut at all):

       \
        \     +------------+
         \   /              \
          \ /                \
           \                  \
          / \                  +
         /   \                 |
        /     \                |
       /       \               |
      +         \              |
       \         \             |
        \         \            |
         \         \           |
          +---------\----------+
                     \
                      \

It seems that each piece will still be convex, so if you cut again, 
you will cut each piece into two pieces again - if you can go through 
both parts. That's not hard:

      \
       \     +------------+
        \   /              \
         \ /                \
          \                  \
         / \                  +
        /   \                 |
    --------------------------------
      /       \               |
     +         \              |
      \         \             |
       \         \            |
        \         \           |
         +---------\----------+
                    \
                     \

Now we've doubled the number of pieces again. To get eight pieces, we 
need to double again, which means we have to cut every one of the 
four pieces. But how can we do that?

Well, the key is to realize that I haven't drawn a cake; a cake is 
three-dimensional. Think of my picture as a top view. Can you see a 
way to cut it again, passing through all four pieces?

A doughnut is different from a cake, because it is not convex. I can't 
see any way to cut it into more than two pieces with one cut, but each 
of those pieces will be concave, and concave pieces can be cut into 
more than two parts with one cut:

             +------------+
            /              \
           /                \
          /                  \
         /       +----+       +
        /       /      \      |
  -----------------------------------
      /       /          \    |
     +       /            \   |
      \     /              \  |
       \   /                \ |
        \ /                  \|
         +                    +

We can cut something into 12 pieces by multiplying it by 2, then 3, 
then 2, rather than by 2 three times as with the cake. Try to find a 
way to cut each of two halves of a doughnut into 3 pieces, and then 
cut every piece again. You will have to use the three-dimensionality 
of the doughnut even more than with the cake! Try making all your cuts 
on an angle, and make at least one of them cut through the existing 
pieces in a way that looks like my picture above.

Incidentally, the problem doesn't say whether you are allowed to 
rearrange the pieces between cuts. If you are, the problem becomes a 
lot easier. It also doesn't say the cuts have to be straight, but I'm 
assuming that is intended.

If you have any further questions, feel free to write back.

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