100 Prisoner riddle-Hard Riddles

100 Prisoner riddle:

100 prisoners are each given a number from 1 to 100 and are placed in a large room. Each prisoner can see the numbers on every other prisoner’s forehead but not their own. The warden then comes in and says:

“I will place a hat on each of your heads. The hat will be either red or blue. If you guess the color of your own hat correctly, you will be freed. If you guess incorrectly, you will be executed. You are not allowed to communicate with each other at all once the process starts. You may, however, plan a strategy beforehand.”

The warden then places the hats on all the prisoners, and they all line up in a row, facing forward (so they can see the hats of the prisoners in front of them but not their own or anyone behind them). The warden asks them to guess the color of their own hat, starting from the back of the line.

What strategy can the prisoners use to maximize their chance of survival?

Hint:

The key is to use parity (even or odd numbers). 🧠

The prisoners can agree beforehand on a strategy based on the parity of the number of red hats they see in front of them. Each prisoner will use the information they see to communicate the total number of red hats they observe, but in a clever way.

This way, they can work together as a group to save as many of them as possible.

Answer:

The Strategy: Use Parity (Even or Odd Numbers)

The prisoners can save everyone except for one, and that one is at the back of the line. Here’s how they can do it:

Before the hats are placed, the prisoners agree on a strategy:
- They agree to base their guesses on the parity (odd or even number) of the red hats they see in front of them.
- The first prisoner (at the back of the line) will announce whether they see an odd or even number of red hats in front of them. This is the key piece of information.
The first prisoner (the one at the back of the line) can see the hats of everyone in front of them.
- If they see an even number of red hats, they say “red” (or any agreed-upon signal for red).
- If they see an odd number of red hats, they say “blue” (or whatever is the agreed-upon signal for blue).
The first prisoner’s guess is purely for information, not necessarily to save themselves. They are essentially communicating the parity of red hats they can see to the rest of the prisoners.
The remaining prisoners (from the second to the last) will know the total number of red hats in front of them by using the first prisoner’s information. Here’s how it works:
- Each prisoner will count the number of red hats in front of them.
- Based on the parity (even or odd) communicated by the first prisoner, they can deduce the color of their own hat.
For example:
- If the first prisoner says “red” (meaning they saw an even number of red hats), and the second prisoner sees an odd number of red hats in front of them, they must have a red hat (because adding their own red hat would make the total number of red hats even).
- If the second prisoner sees an even number of red hats, they must have a blue hat (since their own hat must keep the total number of red hats even).
Each subsequent prisoner continues this process, using the information from all previous guesses to determine their own hat color. They keep track of how many red hats have been identified and use that to figure out if the total number is even or odd.

Outcome:

The first prisoner has a 50/50 chance of survival because they are guessing based on parity, but they communicate valuable information to everyone else.
The remaining 99 prisoners will all be able to deduce their hat color with 100% accuracy based on the information the first prisoner gives and the hats they see in front of them.

Final Answer:

The first prisoner gives a guess based on the parity of red hats they see (odd or even).
The rest of the prisoners use this information to deduce their own hat colors, saving 99 out of 100 prisoners!

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