The 100 Prisoners Problem: A Seemingly Impossible Riddle

The 100 Prisoners Problem: A Seemingly Impossible Riddle

The 100 Prisoners Problem is a classic probability puzzle that seems impossible to solve. It's a riddle that has captivated mathematicians and puzzle enthusiasts for decades, and it's a great example of how seemingly impossible problems can have surprisingly elegant solutions.

The Setup

Imagine 100 prisoners, each assigned a unique number from 1 to 100. They are placed in separate cells. In a room outside the cells, there are 100 boxes, each containing a single prisoner's number. The boxes are arranged randomly, so there's no correlation between the box number and the prisoner's number inside.

The prisoners are given a chance to find their own number. They can enter the room one at a time and open up to 50 boxes. If they find their own number within those 50 tries, they are freed. If they fail, all 100 prisoners are executed.

The Challenge

At first glance, this seems like an impossible task. The odds of any single prisoner finding their number in 50 tries are incredibly low. With 100 boxes, the probability of finding their number on the first try is 1/100. The probability of finding it on the second try is even lower, as the first box they opened will be empty. The probability of failure seems almost certain.

The Surprising Solution

However, there's a strategy that drastically increases the prisoners' chances of survival. It's a strategy based on understanding cycles and using them to their advantage.

Understanding Cycles

Imagine the numbers in the boxes are arranged like this:

1. Box 1: Prisoner 3
2. Box 2: Prisoner 1
3. Box 3: Prisoner 5
4. Box 4: Prisoner 2
5. Box 5: Prisoner 4

In this example, we have a cycle of length 5. Starting from Box 1, we follow the numbers: 3, 5, 4, 2, 1, and back to 3. This cycle has five elements.

The Strategy

Here's how the prisoners can use this strategy to find their own numbers:

1. Start with their own number's box. For example, Prisoner 1 would start with Box 1.
2. Open the box. The box will contain another prisoner's number.
3. Open the box corresponding to the number they just found. For example, if Box 1 contained Prisoner 3's number, they would then open Box 3.
4. Continue following the chain until they find their own number or they have opened 50 boxes.

By following this strategy, the prisoners essentially break down the boxes into cycles. Since each number appears only once, they are guaranteed to find their own number if the cycle length is 50 or less.

Probability of Success

The probability of success using this strategy is surprisingly high. It's greater than 30%! This is a far cry from the near-zero chance of success if they were to open boxes randomly.

Why does this strategy work?

The strategy works because it exploits the fact that the numbers in the boxes are arranged in cycles. By following the chain, the prisoners are guaranteed to eventually reach their own number if the cycle length is 50 or less. The probability of having a cycle longer than 50 is relatively low.

Conclusion

The 100 Prisoners Problem is a fascinating example of how probability can be used to solve seemingly impossible problems. It demonstrates that a well-thought-out strategy can make a significant difference in the outcome, even in situations where the odds seem stacked against you. The next time you encounter a seemingly impossible problem, remember the 100 prisoners and their surprising solution.