# The Riddle That Seems Impossible Even If You Know The Answer

TLDRThis video describes a counterintuitive math riddle involving 100 prisoners trying to find slips of paper with their numbers in boxes. Though it seems impossible, there is a strategy using loops that gives them a 30% chance of success. The video explains this loop strategy in depth, calculating the probability and showing how it works for any number of prisoners. It reveals how linking the prisoners' outcomes together raises their chances tremendously compared to each trying individually. The elegant solution highlights the incredible features of math.

###### Takeaways

- ๐ฒ There is a counterintuitive math riddle where 100 prisoners have a 30% chance of all finding their own numbers if they use a loop strategy.
- ๐ฎโ๐จ If each prisoner searches randomly, they only have a 0.0000000000000000000000008% chance of all succeeding.
- ๐ฏ The prisoners can develop a strategy beforehand to improve their odds.
- ๐คฏ By following loops of numbers between boxes, each prisoner is guaranteed to be on the loop containing their number.
- ๐ตโ๐ซ The probability of success depends on the lengths of the loops formed by the random arrangement.
- ๐ง Longer loops over 50 make it impossible for some prisoners to find their numbers.
- ๐ค With the loop strategy, either all prisoners succeed or most fail together.
- ๐ Each prisoner still has a 50% individual chance, but their outcomes are linked.
- ๐ Having more prisoners barely changes the ~30% success rate.
- ๐ฅณ Brilliant has probability puzzles and courses to develop math problem-solving skills.

###### Q & A

### What is the prisoner's dilemma described in the video?

-There are 100 prisoners numbered 1-100. Slips of paper with their numbers are randomly placed in boxes. Each prisoner gets to open 50 boxes trying to find their number. If all find their numbers, they are freed, otherwise executed.

### What is the probability of success if each prisoner searches randomly?

-Each has a 50% chance of finding their number. So the overall probability is 1/2^100, which is about 1 in 10^30.

### How does the looping strategy work?

-Start with your box. Go to the next number, and keep following the loop until you find your number. This guarantees you find yours if the loop length is <=50.

### What is the probability of success using the loop strategy?

-Around 31%. There is a 69% chance the longest loop is >50, dooming some prisoners. But 31% of arrangements have no loop longer than 50.

### Why does the probability approach a limit with more prisoners?

-The probability of failure equals the area under 1/x from n to 2n. As n increases, this approaches a limit of ln(2), giving a success probability of ~30.7%.

### Can a guard help ensure success or failure?

-Yes, swapping two slips breaks the longest loop, ensuring success. Or, making one long loop dooms the prisoners. But prisoners can renumber boxes to foil this.

### Does the probability decrease substantially with more prisoners?

-No, it stays around 30-31% regardless of the number of prisoners. More prisoners doesn't change the probability much.

### Why is the solution counterintuitive?

-It seems impossible all prisoners could find their numbers by chance. But with this strategy, a large number still have ~30% chance, far higher than random.

### How are the prisoners' outcomes linked?

-By following loops, all prisoners in a loop succeed or fail together. So their probabilities are identical rather than independent.

### What makes this an interesting mathematical puzzle?

-It involves permutations, combinations, probability, and limits. The solution relies on deep mathematical principles despite seeming impossible.

###### Outlines

##### ๐งฉ Introducing the Counterintuitive Riddle

The video introduces a perplexing riddle involving 100 prisoners and a unique challenge where each prisoner must find their own number among 100 boxes within a sealed room, each allowed to open only 50 boxes. The initial setup appears daunting, with a naive strategy offering a virtually impossible chance of success for all prisoners to find their numbers, likened to finding the same grain of sand on Earth. However, the video promises a mathematical strategy that remarkably improves their odds to nearly one in three, a significant leap in probability that challenges intuition and showcases the profound impact of strategic planning and mathematical insight.

##### ๐ The Mathematical Strategy Explained

This segment delves into the ingenious strategy that dramatically increases the prisoners' survival chances. By following a sequence based on the numbers found in each box, prisoners can create a chain that leads them to their own number, leveraging the structure of loops within the arrangement of boxes and slips. This method capitalizes on the mathematical principle that any random arrangement of slips will form closed loops, with the success of each prisoner dependent on the loop's length being 50 or shorter. The explanation unravels the complex probability calculations behind the strategy, revealing a surprising 31% chance of success for all prisoners, a stark contrast to the initial dire predictions.

##### ๐ค Skepticism and Clarifications

The third paragraph addresses common doubts and questions about the strategy, clarifying how each prisoner is indeed guaranteed to find their number by following the loops formed by the box and slip arrangements. It further explores hypothetical scenarios involving a sympathetic or malicious guard's interference, illustrating how prisoners can still maintain or recover their chances of success through strategic adjustments. The discussion emphasizes the robustness of the loop strategy against various potential sabotages, reinforcing the depth and resilience of the mathematical approach.

##### ๐ Probability and Implications

The concluding segment presents the broader implications of the loop strategy, highlighting its effectiveness not just for 100 prisoners but for any number, with the probability of success converging towards a limit around 30.7%. This astonishing consistency across different scales illustrates the counterintuitive nature of probability and strategic thinking. The video wraps up by linking the riddle's insights to broader problem-solving skills, promoting an educational platform for further learning and exploration of mathematical concepts, thereby underscoring the value of critical thinking and mathematical curiosity.

###### Mindmap

###### Keywords

##### ๐ก100 prisoners problem

##### ๐กCounterintuitive

##### ๐กStrategy

##### ๐กProbability

##### ๐กClosed loop

##### ๐กPermutation

##### ๐กFactorial

##### ๐กNatural logarithm

##### ๐กMathematical intuition

##### ๐กInteractive learning

###### Highlights

The introduction provides helpful context about the changing media landscape and the emergence of new communication channels like podcasts.

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The podcast advertising model is attractive because it is native, contextual and unobtrusive.

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Podcasts face challenges in discovery and promotion in a crowded marketplace.

The podcast medium allows for in-depth storytelling, analysis and interviews.

Podcasts have become an influential new information source and mode of entertainment.

Podcasts have opportunities for further monetization and continued growth in the future.

###### Transcripts

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