For two people, this is a quite simple and well known
answer. One
divides, the other
decides.
However, when you have more than two people, it gets trickier. So, picture a round cake of equal height.
- Designate a cutter.
- Make a single cut from to the center of the cake.
- Slowly move the knife clockwise.
- At any point, anyone may say "Now".
- This stops the knife
- Makes the cut
- Gives the cut piece of cake to the person who said "Now"
- Removes the person with cake from saying "Now" again
- If it is the cutter that said now, reassign the cutter.
Repeat until down to 2 people.
This
method works based on the idea that each person wants an
equal part of the cake. Once it gets to 1/N
th of the cake, it is best for people to say 'Now', otherwise
someone else will have a larger piece than 1/N
th of the cake meaning that there will be less
cake for you.
Mathematically, this relies on the Frobenius-Konig theorem.