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.
(In reply to
re(2): define... by Tristan)
I found a problem with my solution.
let X equal the chords' distances from the center and R equal the radius of the cake.
X=R * √π / 2 / 3^(1/4)
So X is about .6733868435*R
The distance from the center to the corners of the pyramid's base should be twice that: about 1.346773687*R. The problem here is that the pyramid extends beyond the cake! This does not mean that there is no solution in the shape I described, but it means that the exact measurements of it are much harder to calculate and beyond my ability. If anyone actually understood my messily explained solution idea, I invite him/her to find the numbers that would make it work.

Posted by Tristan
on 20031213 00:58:27 