The Riddle That Seems Impossible Even If You Know The Answer

Veritasium

30 Jun 202217:45

EducationalLearning

32 Likes 10 Comments

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

😲 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.
😮‍💨 If each prisoner searches randomly, they only have a 0.0000000000000000000000008% chance of all succeeding.
😯 The prisoners can develop a strategy beforehand to improve their odds.
🤯 By following loops of numbers between boxes, each prisoner is guaranteed to be on the loop containing their number.
😵‍💫 The probability of success depends on the lengths of the loops formed by the random arrangement.
🧐 Longer loops over 50 make it impossible for some prisoners to find their numbers.
🤔 With the loop strategy, either all prisoners succeed or most fail together.
😊 Each prisoner still has a 50% individual chance, but their outcomes are linked.
😀 Having more prisoners barely changes the ~30% success rate.
🥳 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

00:00

🧩 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.

05:02

🔍 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.

10:04

🤔 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.

15:05

📈 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

💡100 prisoners problem

The 100 prisoners problem is a mathematical puzzle that involves 100 prisoners, each trying to find their own number among 100 boxes to gain freedom. The script describes a scenario where slips of paper containing each prisoner's number are randomly placed in 100 boxes in a sealed room, and each prisoner is allowed to search 50 boxes. The problem illustrates a counterintuitive solution that significantly increases the prisoners' chances of all finding their numbers, showcasing the fascinating interplay between probability, strategy, and mathematical insight.

💡Counterintuitive

Counterintuitive refers to something that goes against one's initial intuition or common sense. In the context of the video, the solution to the 100 prisoners problem is described as counterintuitive because it defies the logical expectation that the prisoners' chances of success are near impossible. Instead, a specific strategy dramatically improves their odds, highlighting how mathematical principles can yield surprising results.

💡Strategy

Strategy, in the context of this puzzle, involves a premeditated plan the prisoners use to maximize their chances of finding their numbers in the boxes. The script discusses a specific strategy where prisoners follow a sequence from one box to another based on the numbers found inside, creating a 'loop'. This approach significantly increases their collective chance of success, demonstrating the importance of strategy in solving complex problems.

💡Probability

Probability is a measure of the likelihood of a particular outcome or event occurring. In the video, probability is central to understanding the odds of all prisoners finding their numbers using random search versus a structured strategy. The script delves into calculating these probabilities, showing how a seemingly small chance of success can be improved with the right approach.

💡Closed loop

A closed loop in this context refers to a sequence of actions where each step leads to the next in a circular manner until returning to the starting point. The prisoners' strategy relies on finding loops within the arrangement of numbers in boxes. If a prisoner's number is part of a loop that is 50 or fewer steps long, they are guaranteed to find their number within the allowed attempts, illustrating how loop structures impact the outcome of their search.

💡Permutation

Permutation refers to the arrangement of all elements of a set in a specific order. The script explains how the total number of permutations of arranging 100 slips in 100 boxes is calculated as 100 factorial. This concept is crucial for understanding the vast number of possible configurations the prisoners face, and the calculation of the odds of certain loops forming within these permutations.

💡Factorial

Factorial, denoted by an exclamation mark (!), refers to the product of an integer and all the integers below it, down to 1. For example, 100 factorial (100!) is the product of all numbers from 1 to 100. This mathematical operation is used in the video to calculate the total number of ways to arrange the slips in boxes, providing a foundation for understanding the probability calculations presented.

💡Natural logarithm

The natural logarithm is a logarithm in base e, where e is approximately equal to 2.71828. It's used in the video to calculate the probability of failure as the area under the curve of 1/x, leading to the solution that involves the natural logarithm of two. This illustrates how integral calculus and logarithmic functions are applied to derive probabilities in complex scenarios.

💡Mathematical intuition

Mathematical intuition refers to the instinctive understanding or insight into mathematical concepts and solutions. The script mentions that even the person who proposed the 100 prisoners problem did not initially see the counterintuitive solution, highlighting the challenge of overcoming intuitive misconceptions with rigorous mathematical reasoning.

💡Interactive learning

Interactive learning is mentioned at the end of the script as a method to engage with and understand complex problems like the 100 prisoners problem. It suggests using platforms like Brilliant to explore similar puzzles through engaging, interactive lessons. This approach to learning emphasizes the importance of actively engaging with material to develop a deeper understanding of mathematical concepts.

Highlights

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Transcripts

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Takeaways

Q & A

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

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

How does the looping strategy work?

What is the probability of success using the loop strategy?

Why does the probability approach a limit with more prisoners?

Can a guard help ensure success or failure?

Does the probability decrease substantially with more prisoners?

Why is the solution counterintuitive?

How are the prisoners' outcomes linked?

What makes this an interesting mathematical puzzle?