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/Nth of the cake, it is best for people to say 'Now', otherwise someone else will have a larger piece than 1/Nth of the cake meaning that there will be less cake for you.

Mathematically, this relies on the Frobenius-Konig theorem.