3 Prisoners Hat Guessing Game with Passing (Hamming Strategy)

Scenario

Three prisoners are assigned either a Red or Blue hat independently and with equal probability $1/2$. Each prisoner can see the other two prisoners' hats, but cannot see their own. All three prisoners simultaneously make a call: each may either guess their own color (Red or Blue) or choose to PASS. The team wins if at least one prisoner guesses correctly and no prisoner guesses incorrectly. If anyone guesses wrong, or if all three pass, the team loses.

Your Task

1. Prove that in any valid strategy, an individual guess has a conditional error rate of $50\%$.
2. Explain why maximizing team win probability requires concentrating all incorrect guesses into as few joint hat configurations as possible.
3. State the optimal single-error Hamming covering strategy and prove that it achieves a $75\%$ win rate.

Constraints

• Simultaneous declarations; zero communication.

Examples In configuration (Red, Red, Blue), the prisoner seeing two Red hats guesses Blue and is correct. The other two see (Red, Blue) and pass. Team wins!