The solution is simple….?

The pattern of hats on three heads can be 8 different permutations:

red, red, red
red, red, blue
red, blue, red
red, blue, blue
blue, red, red
blue, red, blue
blue, blue, red
blue, blue, blue

The trick is to “see” that 2 of these 8 permutations have a distinguishing characteristic: they involve all three hats being the same color.

Think about it ….. 1/4 of the time, the hats are all the same. 3/4 of the time they are NOT all the same ….. 1 is red and 2 are blue, or 2 are red and 1 is blue.

So if the players all make the assumption that the colors are split 1 and 2, and vote appropriately, they will win 3/4 of the time, by following this strategy:

If you see 2 hats of the same color, make a “guess” that your own hat is of the opposite color. But if you see hats of opposite colors, write “pass” on your slate.

Welcome to the 3-Hat Problem!

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.