I was sitting down with Stefanie one day to share a round cake (our birthdays are only two weeks apart). "This is easy enough," I said, "one cut right through the middle divides the cake into two equal pieces."

Then, two more people showed up, but I was undaunted. Two straight cuts will divide the cake into four equal parts, I thought.

Then, I saw another car pulling up. I remembered that three straight lines can divide a circle into at most seven parts, but I was unsure if that could be done so that all the pieces are equal (in volume, not necessarily in shape).

How can I use three straight cuts to divide our cake into all equal parts and accomodate the greatest number of people?

Note: since Stefanie spent so much time decorating the cake, I don't want to rearrange the pieces when I cut them.

Consider this as a 2-D problem to make 7 equal-sized pieces:

The first cut (a chord of the circular top of the cake) is going to have to cut the circle into a ratio of areas of 3:4. That's easy enough, and there's a unique distance from the center that will do this. Rotationally it doesn't matter where you do this, so you can't mess up here.

The next cut has to cut that smaller previous piece in a 1:2 ratio and the larger previous piece in a 2:2 ratio (i.e., in half--left unreduced to emphasize the double-size nature of the pieces left). That should also be possible, given that there are two degrees of freedom in placing that second chord, though it will be a little tricky figuring out where.

But then the third cut must divide each of three double-sized pieces in half. Unless the pieces are fortunately placed (and I doubt that they would be from the constraints imposed by the previous step), you wouldn't be able to divide all three remaining pieces in half. That would be accomplishing 3 things with just the 2 degrees of freedom in choosing the chord.

Edited on December 4, 2003, 9:07 am
Edited on December 4, 2003, 9:09 am

Posted by Charlie on 2003-12-04 09:07:02