# Proof by induction in a game with water balloons

Consider the following description of a game. There are $$n$$ people playing, one of whom leads the game. They are playing on a playing field with no obstacles. Everyone carries one water balloon.

Everyone walks around the playing field until the game leader yells “stop”. Assume that the leader will yell stop only when everyone is in a position where they have a unique closest neighbour.

Next, the leader yells “throw” and everyone throws their water balloon at their nearest neighbour (there is exactly one, by the assumption in the previous point).

A survivor is anyone who is still dry at the end of the game (assuming everyone has perfect aim and the water balloons are designed so that they get only their intended target wet).

Answer the following questions to prove that when the game is played with an odd number of players, there will always be at least one survivor.

(a) Define a statement $$P(n)$$ that will enable you to prove that there is always at least one survivor when the game is played with an odd number of people. Design your statement $$P(n)$$ so that you can prove it true $$∀ n ∈ ℕ$$.

(b) Prove $$\forall n \in \mathbb{N}, P(n)$$. Hint: Try to figure out how to reduce the number of people playing the game.

Hey guys, a little help would be appreciated. I think I am capable of doing part b) if i know part a)

$P(n)$ is simply the statement that if the game is played with $2n+1$ players, there is a survivor. HINT for (b): Consider the two players who are closest together when the leader says stop.