Prisoners and Hats Puzzle

Problem:

Four prisoners are arrested for a crime, but the jail is full and the jailer has nowhere to put them. He eventually comes up with the solution of giving them a puzzle so if they succeed they can go free but if they fail they are executed.

The jailer puts three of the men sitting in a line. The fourth man is put behind a screen (or in a separate room). He gives all four men party hats. The jailer explains that there are two black and two white hats; that each prisoner is wearing one of the hats; and that each of the prisoners is only to see the hats in front of them but not on themselves or behind. The fourth man behind the screen can’t see or be seen by any other prisoner. No communication between the prisoners is allowed.
If any prisoner can figure out and say to the jailer what color hat he has on his head all four prisoners go free. If any prisoner suggests an incorrect answer, all four prisoners are executed. The puzzle is to find how the prisoners can escape, regardless of how the jailer distributes the hats.

prisoners and hat puzzle

Solution:

Prisoner A and B are in the same situation – they have no information to help them determine their hat colour so they can’t answer. C and D realise this.

Prisoner D can see both B and C’s hats. If B and C had the same colour hat then this would let D know that he must have the other colour.

When the time is nearly up, or maybe before, C realises that D isn’t going to answer because he can’t. C realises that his hat must be different to B’s otherwise D would have answered. C therefore concludes that he has a black hat because he can see B’s white one.

5 Thoughts on “Prisoners and Hats Puzzle

  1. Niranjan Singh on February 7, 2014 at 3:53 pm said:

    what if C and D have same colored hats.

    • Rohit Kumar on April 5, 2016 at 11:08 am said:

      doesn’t matter answer will be same … if c and d black caps then also d will not reply so c can easily assume that he is wearing black cap or viceversa,

  2. Ron McMullin on March 22, 2014 at 12:03 am said:

    That wouldn't matter. The logic of C still follows.

  3. Utkarsh verma on November 25, 2015 at 9:46 am said:

    What about a and d?

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