# Optimizing Prisoner Strategy for the Hat Riddle: A Simple Solution

• collinsmark
In summary, the prisoners are given a task (to guess the color of their own hat) the night before and only those who guess correctly are set free.
Here is my solution. It's equivalent to @Vanadium 50's solution, but I'll go into more detail with emphasis on it being simple -- mentally speaking -- in terms of what each prisoner must do while in line.

So, Crusty explains that each prisoner must be observant and keep track of two things, the "running parity" and their own special "personal parity."

"The first thing you need to figure out," he says, "is to determine whether the number of white hats in front of you is even or odd. Use whatever method you want; count them, pair them up, whatever. But you do need to figure this out before the guessing gets started. You can do this in conjunction with the warden putting hats on people, so there's really no rush -- you'll have plenty of time. Call this evenness or oddness your 'personal parity.' It is either even or odd. Your personal parity is even if there are an even number of white hats in front of you. Your personal parity is odd if you see an odd number of white hats. You only need to figure this out once and it never changes. By the way, zero counts as even, so if there are zero white hats in front of you, your personal parity is even.

"Next, you'll also have to pay attention to something we'll call the 'running parity.' The running parity starts out even. Every time you hear somebody behind you guess 'white,' the running parity toggles. It starts out as even. The first time somebody says 'white,' the running parity changes to odd. The next time somebody says 'white' it changes back to even. And so on.

"When somebody guesses 'black' the running parity remains what it is. If it was even it stays even. If it was odd it stays odd. Only 'white' guesses make it change from even to odd or odd to even.

"So here is what you do when it's your turn: Compare the running parity to your personal parity. If they are the same, you have a black hat, so say 'black.' If they are different you have a white hat, so say 'white.'"​

The way I worded it above is equivalent to the first guy -- the guy in the back of the line -- saying "white" if the number of hats he sees is odd, and "black" if he sees an even number of white hats (same as @Vanadium 50's solution). I phrased it the way I did so the guy in back doesn't need to follow any special rules. But it's equivalent: he has a 50/50 chance of making it out alive. All the other prisoners have 100% chance of being set free if they follow the plan.

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<h2>1. What is the Hat Riddle and why is it important to optimize prisoner strategy?</h2><p>The Hat Riddle is a classic logic puzzle that involves prisoners wearing either a black or white hat and trying to guess the color of their own hat without being able to communicate with each other. It is important to optimize prisoner strategy because it can potentially lead to a successful escape and highlights the importance of strategic thinking and cooperation.</p><h2>2. What is the simple solution to optimizing prisoner strategy for the Hat Riddle?</h2><p>The simple solution involves the first prisoner counting the number of black hats and then guessing the color of their own hat based on whether the number is even or odd. The remaining prisoners then use this information to determine the color of their own hat. This strategy has a 50% chance of success.</p><h2>3. Are there any other strategies that can be used to optimize prisoner strategy for the Hat Riddle?</h2><p>Yes, there are other more complex strategies that can be used, such as using a predetermined code or signaling system. However, the simple solution is the most efficient and has the highest chance of success.</p><h2>4. How does the number of prisoners affect the success rate of the simple solution?</h2><p>The success rate of the simple solution remains at 50% regardless of the number of prisoners. However, the more prisoners there are, the more difficult it becomes to execute the strategy effectively without any errors.</p><h2>5. Can the simple solution be applied to other similar logic puzzles?</h2><p>Yes, the simple solution can be applied to other similar logic puzzles that involve a group of individuals trying to determine a specific piece of information without being able to communicate with each other. It highlights the importance of teamwork and strategic thinking in problem-solving scenarios.</p>

## 1. What is the Hat Riddle and why is it important to optimize prisoner strategy?

The Hat Riddle is a classic logic puzzle that involves prisoners wearing either a black or white hat and trying to guess the color of their own hat without being able to communicate with each other. It is important to optimize prisoner strategy because it can potentially lead to a successful escape and highlights the importance of strategic thinking and cooperation.

## 2. What is the simple solution to optimizing prisoner strategy for the Hat Riddle?

The simple solution involves the first prisoner counting the number of black hats and then guessing the color of their own hat based on whether the number is even or odd. The remaining prisoners then use this information to determine the color of their own hat. This strategy has a 50% chance of success.

## 3. Are there any other strategies that can be used to optimize prisoner strategy for the Hat Riddle?

Yes, there are other more complex strategies that can be used, such as using a predetermined code or signaling system. However, the simple solution is the most efficient and has the highest chance of success.

## 4. How does the number of prisoners affect the success rate of the simple solution?

The success rate of the simple solution remains at 50% regardless of the number of prisoners. However, the more prisoners there are, the more difficult it becomes to execute the strategy effectively without any errors.

## 5. Can the simple solution be applied to other similar logic puzzles?

Yes, the simple solution can be applied to other similar logic puzzles that involve a group of individuals trying to determine a specific piece of information without being able to communicate with each other. It highlights the importance of teamwork and strategic thinking in problem-solving scenarios.

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