Topic: Induction.

For this assignment, you will present an inductive proof for a variation on a classic logic puzzle. In your writeup, make sure to clearly label your base case$($s$),$ your inductive hypothesis, and your induction step.

The puzzle: $50$ friends sit down to play a game. One chosen "referee" sticks either a yellow or a blue dot to each player's forehead. Players cannot see their own dots, but they are able to see all other players' dots. The players are not allowed to speak. Each round, any player who knows their dot color may raise their hand and "win" the game.

Prior to the first round, the referee announces that there is at least one blue dot on a player's forehead.

There is inductive logical reasoning that leads to the following outcome: If there are X players with blue dots, then all of those players correctly raise their hands on round X$.$ The remaining players correctly raise their hands on round X$+1.$

Your writeup should be a complete inductive proof. Make sure you clearly state:

Your base case$($s$)$

Your inductive hypothesis

Where you invoke your inductive hypothesis

The key insight in your induction step