Veritasium - The Riddle That Seems Impossible Even If You Know The Answer
The problem involves 100 prisoners who must each find their own number in one of 100 boxes. Each prisoner can open 50 boxes, and they must leave the room as they found it without communicating with others. If all prisoners find their numbers, they are freed; if one fails, they are executed. Randomly searching gives each prisoner a 50% chance of success, making the collective probability of success astronomically low. However, a specific strategy can increase their chances to nearly one. This strategy involves a mathematical approach that improves their odds by nearly 30 orders of magnitude compared to random searching.
Key Points:
- 100 prisoners must find their numbers in 100 boxes, each opening 50 boxes.
- Random searching gives a near-zero probability of success for all prisoners.
- A specific strategy can increase the probability of success to nearly one.
- The strategy improves odds by nearly 30 orders of magnitude over random chance.
- The problem highlights the power of mathematical strategies in problem-solving.
Details:
1. 🔍 The Prisoner Riddle Setup
- The riddle is described as counterintuitive, suggesting that even with the correct answer, it appears incorrect.
- Involves 100 prisoners, each numbered from 1 to 100.
- Prisoners are faced with a challenge involving a series of boxes, each containing a slip with a number, and must find their own number to survive.
- The objective is for every prisoner to find their number in a limited number of attempts to ensure their survival.
2. 🚪 The Rules of the Game
- 100 unique numbers are distributed randomly into 100 boxes within a sealed room.
- Each prisoner is allowed to enter the room individually and open any 50 of the 100 boxes to find their number.
- Prisoners must leave the room exactly as they found it with no means of communication with others.
3. 🎲 The High Stakes
- If all 100 prisoners find their own number during their turn, they will all be freed, but if one fails, they will all be executed.
- Prisoners can strategize before any of them enters the room, suggesting the importance of a collective and effective strategy.
- The scenario emphasizes the critical need for a fail-proof method, as a single failure results in execution, highlighting the high stakes involved.
- One potential strategy involves using a loop-based method, where each prisoner follows a sequence based on the number they find, increasing their chances of success.
4. 📉 The Probability of Failure
- The problem involves 100 prisoners and 100 boxes, each containing a number. Each prisoner must find their own number by opening up to 50 boxes.
- If each prisoner searches randomly, the probability of all prisoners finding their numbers is extremely low, calculated at about 1 in 10^30.
- A more strategic approach, such as following a fixed sequence or pattern, significantly improves the probability of success.
- This strategic method involves each prisoner starting with the box of their own number and then following the number found to open the next box, creating a loop or cycle.
- The strategic approach boosts the success probability to over 30% compared to random searching.
- This improvement highlights the impact of strategy on outcomes even in seemingly random scenarios.
5. 🔢 The Mathematical Strategy
- The probability of two individuals randomly selecting the same grain of sand from all beaches and deserts on Earth is astronomically low, highlighting the challenge of succeeding without a strategy.
- Implementing the correct mathematical strategy can increase the probability of success to nearly 1 in 3, demonstrating an improvement over random chance by nearly 30 orders of magnitude.
- The strategy involves specific steps that leverage mathematical principles to systematically increase the likelihood of success, though these steps are not detailed in this summary.
- Understanding and applying this strategy transforms an otherwise insurmountable task into a manageable one, emphasizing the power of strategic thinking.