Three Hats and One Agreement

Issue 19 Core Published 25 Sep 2026 By The Chalkboard Desk

The problem

Three players each get a red or blue hat placed on their head. Each sees the other two hats but not their own. Simultaneously each player either guesses their own colour or passes; they agree on a strategy beforehand and win if at least one guesses correctly and nobody guesses wrongly. What is the best possible winning probability, and what strategy achieves it?

Hats are assigned independently and uniformly at random, so there are equally likely configurations.
Guesses are simultaneous; no player sees another's decision.
A single wrong guess loses the round even if someone else guessed correctly.

Take a minute

Reveal hint 1

Count configurations rather than players: you need a rule that is correct on of the hat assignments and silent on the other .

Reveal hint 2

Think of a guess as a claim about the parity of the total number of red hats.

Solution

Answer, achieved by a parity strategy.

Derivation

Name the configurations.
Design the strategy.
Check a configuration with an even number of reds.
Count the winning configurations.
Compute the probability.
Why it is optimal.

Margin notes

Write , , and so on — equally likely assignments, each with probability .
Player assumes the number of red hats is even; players and assume it is odd. Each player, seeing the other two hats, guesses the colour that would make the total reds have their assumed parity.
Take (three reds, odd). Player sees two reds and, assuming an even total, guesses blue — wrong. Players and assume odd, see two reds, and guess red — both correct. The round is lost because one guess was wrong.
The players win exactly when the true number of reds disagrees with *one* of the assumptions — which happens for every configuration except and . That leaves of .
.
Every deterministic strategy makes at most correct configurations win, because on each of the assignments at least one player's assumption is violated. Mixing strategies cannot beat the best fixed one, so is the maximum.