Can you solve the prisoner boxes riddle? - Yossi Elran | Summary and Q&A

Summary

In this video, a band finds themselves tied up in a soundproof room on the day of their big concert. Their instruments have been placed randomly in ten boxes outside the room. Each band member can look inside five boxes before being taken back to the tour bus, without being able to communicate with the others. The chances of each musician finding their own instrument through random guessing are very low. However, the drummer comes up with a strategy that has a more than 35% chance of success.

Questions & Answers

Q: What is the drummer's strategy?

The drummer's strategy is for each band member to first open the box with the picture of their instrument. If their instrument is inside, they are done. If not, they should look at what's in the box and then open the box with that picture on it. They should continue this process until they find their instrument.

Q: Why does the drummer's strategy work?

The drummer's strategy works because each musician follows a linked sequence that starts with the box whose outside matches their instrument and ends with the box actually containing it. This creates a loop. By starting with the box with the picture of their instrument, each musician restricts their search to the loop that contains their instrument. There are decent odds, about 35%, that all of the loops will be of length five or less.

Q: How are the odds of success calculated?

For a simplified case with four instruments and no more than two guesses allowed for each musician, the odds of failure (needing to open three or four boxes) and success (finding the instrument within two guesses) can be calculated. There are six distinct four-box loops and eight distinct three-box loops. By visualizing the loops using squares and triangles, the number of possible combinations leading to failure and success can be determined. In this case, the odds of success are about 35%.

Q: Does the computational strategy work for any number of musicians?

Yes, the computational strategy can work for any even number of musicians. However, if a shortcut is desired, a general equation can be used. By plugging in the number of musicians, such as ten, the odds of success can be calculated. For ten musicians, the odds are about 35%. As the number of musicians increases, the odds approach about 30%. While not a guarantee, with some luck, it is far from hopeless.

Q: What are the odds of success if there were 1,000 or 1,000,000 musicians?

As the number of musicians increases, the odds of success approach about 30%. For example, if there were 1,000 or 1,000,000 musicians, the odds would still be around 30%. While it is not a guarantee, there is still a significant chance of success with a bit of musician's luck.

Takeaways

The drummer's strategy of following a linked sequence of boxes that starts with the box matching their instrument and ends with the box containing their instrument greatly increases the odds of success. By restricting their search to a loop, the chances of finding the instrument within five guesses are about 35%. The computational strategy and equation provided can be used to calculate the odds of success for different numbers of musicians. With a bit of luck, even with a large number of musicians, the odds can still be favorable.