The 3 Hat Problem is the outer layer of an onion. Once you solve is, try the 7-Hat version. Then move on to the n-Hat versions. It is very counter-intuitive, in that, the more players, the greater the chance of winning!
3 players can win 3/4 of the time. 7 players, 7/8 of the time (if they are able to analyze the problem, and agree on a winning strategy before the hats are placed on their heads.)
2^n – 1 players can win (2^n – 1) / 2^n of the time, if they can handle the math needed to reach a solution.
But the 3-Hat situation is unique, in that it can be understood, and won, without any prior discussion or agreement.
EST's 3HP 