Ten prisoners, ten hats, and one guess each — save yourselves with silence
Ten prisoners stand in a line, each seeing every hat in front of them but not their own or anyone behind them. Hats are red or blue, assigned in any pattern. Starting from the back, each prisoner says only 'red' or 'blue' aloud, in order, and is freed if it matches their own hat. With no other communication allowed, what strategy guarantees at least nine of them go free?
Reveal the answer
The prisoner at the back counts the red hats they can see and says 'red' if that count is odd, 'blue' if even, sacrificing their own 50/50 guess to broadcast one bit of information. Every prisoner after that can work out their own hat colour by comparing the running parity to the hats they've seen and answers they've heard, guaranteeing all nine remaining prisoners answer correctly. It's a classic 'induction puzzle,' where each answer narrows down what everyone still to come can safely conclude.