# Guess the color of the cap

I have 3 persons which either wear a white or a black cap. They can only see the color of the other caps, but not their own. White and black caps are eqally likely. As a team, they play a game of guessing their own cap color. If they will win, all of them have to guess correctly their own cap color. Once the game begins, they cannot communicate the color of the other two caps.

Now my interesting question: What is a good strategy for the 3 persons such that with 75% probability all answer correctly? (if the strategy should be made before the caps are donned)

The hard thing is that the players cannot communicate anything with each other once the have the caps.

• "If they win, all of them have to guess correctly" That's really unclear. It sounds like they are playing some other game, and if they happen to win that game, then they have to guess their hat colour. That's probably not what you meant. Also, how are they guessing? All at the same time? One by one in a dedicated order? One by one but they can freely choose which order as part of their strategy? – Arthur Mar 1 at 5:56
• The thing is: one is free how they guess (which order etc). This, one should determine in the strategy. There are really no further rules than I described... – JohnD Mar 1 at 6:03
• Hint: every time at least two hats will be the same color and at least one person will see the other two people wearing the same color hat. – fleablood Mar 1 at 6:48
• Oh, and please remember to accept my answer. Don't just say "thank you". – Parcly Taxel Mar 1 at 9:43
• I thank you AND I accept your answer ;) Thanks a lot! :) – JohnD Mar 1 at 9:46

Hint: Thinking about the different possibilities of hat distribution, there is one thing which happens $$75\%$$ of the time. Find a strategy they can follow in that case, and ignore the remaining $$25\%$$ of cases.