Hard
logic-puzzlesdeductionprisoners-puzzle

The Prisoners and Hats Logic Puzzle Explained

Problem Statement

Three prisoners are in a line, each wearing either a black or white hat. They can see the hats in front of them but not their own or behind. The back prisoner can see both others, the middle sees one, the front sees none. They must guess their own hat color. If at least one guesses correctly and none guess incorrectly, they go free. No communication is allowed. The back prisoner says 'I don't know.' The middle prisoner says 'I don't know.' The front prisoner says 'I know!' and guesses correctly. How?

Answer

The front prisoner's hat is white

Step-by-Step Explanation

  1. 1

    Let's denote: B = back, M = middle, F = front prisoner.

  2. 2

    Setup: B can see M and F. M can see F. F sees no one.

  3. 3

    Key insight: The back prisoner's 'I don't know' gives information.

  4. 4

    If B saw two hats of the same color, B would know their own hat is the opposite color (since there are only 3 hats total, at least one of each color must exist for the puzzle to work — actually, this assumption varies by version).

  5. 5

    Wait, let me reconsider the standard version:

  6. 6

    Standard version: There are 2 black hats and 3 white hats total (or 3 black and 2 white).

  7. 7

    If B sees 2 black hats, B knows their hat is white (only 2 black exist).

  8. 8

    B says 'I don't know' → B doesn't see 2 black hats → M and F are not both black.

  9. 9

    M hears this and sees F. If F were black, M would know M is white (since not both black).

  10. 10

    M says 'I don't know' → F is not black → F must be white.

  11. 11

    F hears both 'I don't know' statements and deduces: F is white.

  12. 12

    This is the classic solution using logical deduction from negative information.