Three Hats Puzzle - Solution
The Puzzle:
The color of each hat is purely random. All hats could be green, or blue, or 1 blue and 2 green, or 2 blue and 1 green.
They need to guess their own hat color by writing it on a piece of paper, or they can write "pass".
They cannot communicate with each other in any way once the game starts. But they can have a strategy meeting before the game.
If at least one of them guesses correctly they win $50,000 each, but if anyone guess incorrectly they all get nothing.
What strategy would give the best chance of success?
(Hint: 100% chance of success is not possible.)
Our Solution:Simple strategy: Elect one person to be the guesser, the other two pass. The guesser chooses randomly "green" or "blue". This gives them a 50% chance of winning.
Better strategy: If you see two blue or two green hats, then write down the opposite color, otherwise write down "pass".
It works like this ("−" means "pass"):
Hats: GGG, Guess: BBB, Result: Lose
Hats: GGB, Guess: −−B, Result: Win
Hats: GBG, Guess: −B−, Result: Win
Hats: GBB, Guess: G−−, Result: Win
Hats: BGG, Guess: B−−, Result: Win
Hats: BGB, Guess: −G−, Result: Win
Hats: BBG, Guess: −−G, Result: Win
Hats: BBB, Guess: GGG, Result: Lose
Result: 75% chance of winning!
Better strategy: If you see two blue or two green hats, then write down the opposite color, otherwise write down "pass".
It works like this ("−" means "pass"):
Hats: GGG, Guess: BBB, Result: Lose
Hats: GGB, Guess: −−B, Result: Win
Hats: GBG, Guess: −B−, Result: Win
Hats: GBB, Guess: G−−, Result: Win
Hats: BGG, Guess: B−−, Result: Win
Hats: BGB, Guess: −G−, Result: Win
Hats: BBG, Guess: −−G, Result: Win
Hats: BBB, Guess: GGG, Result: Lose
Result: 75% chance of winning!