"A crowd of at least two people stands in a room and each one holds a cake. At the sound of a whistle, each person throws their cake at the person closest to them. If the number of people in the crowd is odd, then there is someone who does not get a cake thrown at them. Prove this. Assume that all the distances between pairs of people are distinct."
So far, I think I should use n = 3 as the base case (not sure about any others) and then do the inductive step with n + 2 from the base case (all odd numbers I believe). My only problem is, how would I prove things like a cake not being thrown at a person and how could I apply this for odd numbers. And also, if I need one, how would I state the induction hypotheses?
Any help would be highly appreciated!