The "4 prisoners problem" is a classic logic puzzle that highlights the power of probability and strategy. It presents a scenario where four prisoners, labeled A, B, C, and D, are each given a hat. They can see the hats of the prisoners in front of them but not their own. The hats are either black or white, and the prisoners are told that at least one hat is black.
The goal is for one prisoner to correctly guess the color of their own hat, and the prisoners can communicate with each other before the guessing begins.
Here's the problem:
- Prisoner A is at the back of the line and can see the hats of B, C, and D.
- Prisoner B can see the hats of C and D.
- Prisoner C can see the hat of D.
- Prisoner D cannot see any hats.
The Solution:
The prisoners can use a strategy based on the information they have. Here's how it works:
- Prisoner A observes the hats in front of them. If they see at least one black hat, they can deduce that their own hat must be black. Since they know at least one hat is black, and they see one in front of them, their own hat must be the remaining black hat.
- If Prisoner A doesn't say anything after a reasonable amount of time, Prisoner B knows that A did not see a black hat in front of them. This means that Prisoner B's hat must be black, as it is the only remaining black hat.
- If Prisoner B doesn't say anything, Prisoner C can deduce that their hat must be black. They know that at least one hat is black, and since A and B didn't say anything, the only remaining possibility is that their hat is black.
Why this works:
- The prisoners use the information provided (at least one black hat) and their observations to deduce the color of their own hat.
- The strategy relies on the fact that each prisoner can infer information from the silence of the prisoners ahead of them.
Example:
Let's say the hats are distributed as follows:
- A: Black
- B: White
- C: Black
- D: White
- Prisoner A sees two black hats (C and D) and immediately announces that their hat is black.
- Prisoner B sees one black hat (C) and knows that A would have spoken up if they saw a black hat. Since A is silent, B knows their own hat is black.
- Prisoner C sees one white hat (D) and knows that A and B would have spoken up if they saw a black hat. Since both are silent, C knows their own hat is black.
- Prisoner D, unable to see any hats, cannot deduce the color of their own hat.
Conclusion:
The 4 prisoners problem demonstrates how logical deduction and communication can be used to solve complex puzzles. By carefully observing the information available and using the power of silence, the prisoners can increase their chances of correctly guessing the color of their own hat.
