Proving Existence of a Survivor in a Discrete Math Problem | Odd n Case

[zohapmkoftid]

Homework Statement

Suppose n > 1 people are positioned in a feld, so that each has a unique nearest neighbour. Suppose further that each person has a ball that is thrown at the nearest neighbour. A survivor is a person that is not hit by a ball. Prove that if n is odd, then there is at least one person surviving.

Homework Equations

The Attempt at a Solution

I try to prove by mathematical induction.
When n = 3
Let P₁, P₂, P₃ be the 3 people and d₁₂, d₂₃, d₁₃ be the distance between P₁ and P₂, P₂ and P₃, P₁ and P₃ respectively
Then I consider the different cases such as d₁₂ < d₂₃ and d₁₃ > d₁₂...etc
Under each situation, I can successfully prove that there is one person surviving.

After that, I assume the statement is true for n = 2k + 1
However, I have no idea of how to continue the proof

 

[Office_Shredder]

If there are 2k+1 people, it might help to consider a subset of 2k-1 of them and use your inductive hypothesis