Three people enter a room and have a green or blue hat placed on their head. They cannot see their own hat, but can see the other hats.
The color of each hat is purely random. They could all be green, or blue, or any combination of green and blue.
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 is the best strategy?
The given solution offers a 75% chance at winning. Here is a 100% method:
Before the game the people agree that if they see a blue and a green hat, they will write pass. If they see a match, they will pause. If everyone pauses, they must all have the same color and they can all safely guess that color. If two people immediately write pass, the third knows to guess the opposite of what the other two are wearing.
I came up with a seemingly similar solution.
The three would decide to line up in any order. We would label the person in the back as the 3rd, the middle 2nd, and the front 1st.
Assuming you can change the writing format when writing pass, you could write it in uppercase / capitalize the P if you see a matching pair or in lowercase if there's a diversity. Otherwise you just have to find a subtle way of communicating like pausing. Or does this still violate the "no communicating in any way" rule? You won't talk with each other, make noises, do hand gestures, face expressions but make a pause (like for say 5 seconds).
The 3rd person would take his move based on the two hats he is seeing, if a diversity is present he would write pass immediately, if a pair of color exists instead he would make a pause before writing down pass. By 3rd's action the 2nd person would be aware if a diversity exists or not, he would just say the opposite color or the same color of 1st' hat in order to win.
How would they know if the two other two wrote capital P or not?
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment