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.