Dividing A Cake - A Math Puzzle

Date: 12/20/98 at 19:26:16
From: jim
Subject: Dimensions/algebra/cake cutting

You have a nine-inch-square cake. It is a two-layer cake. Each layer is 
one and one-half inches high. Each layer is covered with 1/4-inch-thick 
frosting and so are the sides.  There is no frosting on the bottom.  
Using straight knife cuts, how can you divide the cake into 13 pieces 
so that each piece has exactly the same amount of cake and frosting?  

After pieces are cut they may be put back together, but you may not 
remove any cake or frosting.

Date: 12/21/98 at 13:23:03
From: Doctor Wilkinson
Subject: Re: Dimensions/algebra/cake cutting

This is a very nice puzzle. In case you want to work on it a little 
longer on your own, I'll give you a couple of hints. If you get stuck 
or you just want the answer, let me know.

13 is too big; it's hard to visualize the problem with such a large 
number of pieces. Try some smaller number like 5 (4 or 2 would be too 
easy).

If the cake were circular instead of square, it would certainly be 
easy. You would just mark off points on the circumference that would 
divide it into 5 equal parts, and cut from the center of the circle to 
these points, right?

Now see if you can do something similar with a square cake.

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

Date: 12/21/98 at 20:38:11
From: Anonymous
Subject: Re: Dimensions/algebra/cake cutting

Thank you very much for your response.  These hints are very helpful. 
I think that I know how to go about solving the problem, but could you 
please send me an explained solution?  

Thank you very much,
Jim

Date: 12/23/98 at 11:21:37
From: Doctor Wilkinson
Subject: Re: Dimensions/algebra/cake cutting

The hint I gave you was really a very good one. You just need to be 
bold and follow it exactly. If you divide the circumference of the cake 
into 13 equal parts and then make your cuts from the center of the cake 
to each of the 13 points on the circumference, all the pieces will have 
equal amounts of both cake and frosting!

To see this, notice first that the amount of cake in a piece is just 
the area of the top of the piece times the thickness of the cake. The 
amount of frosting from the top is just the area of the top times the 
thickness of the frosting, and the amount of frosting from the side is 
just the length of the part of the piece on the edge of the cake times 
the thickness of the cake times the thickness of the frosting. Now if 
you divide the circumference into 13 equal parts, you've taken care of 
the frosting on the side, and you just need to be sure that the pieces 
have equal area to take care of the cake and the frosting on the top.

There are two kinds of pieces of cake if you divide it as I suggested: 
triangular wedges, and pieces that go around a corner of the cake. For 
the triangular pieces, the area is one half the base times the height.  
The height is just the distance from the center of the cake to the 
edge, which is the same for all the pieces, and the base is one of 
those equal divisions of the circumference, so that's the same for all 
the triangular pieces also. I'll let you think about the case of the 
around-the-corner pieces, but they work too.

Here's a picture:

     

- Doctors Wilkinson and Sarah, The Math Forum
  http://mathforum.org/dr.math/