Blue forehead room solution | Puzzles | Math for fun and glory | Khan Academy

TL;DR

A group of logicians with blue foreheads use logical reasoning to determine when to leave a room.

Transcript

So we had the hundred logicians. All of their foreheads were painted blue. And before they entered the room, they were told that at least one of you hundred logicians has your forehead painted blue. And then every time that they turned on the lights, so that they could see each other, they said OK, once you've determined that you have a blue forehe... Read More

Key Insights

• 😒 Logicians use logical reasoning to determine when to leave based on the number of logicians with blue foreheads.
• 💁 The puzzle highlights the importance of considering multiple scenarios and making deductions based on incomplete information.
• 🥺 The increase in the number of logicians helps eliminate possibilities and leads them to the correct conclusion.

Questions & Answers

Q: How does the logician determine if they have a blue forehead?

By observing the number of logicians with blue foreheads in each round and realizing they would have left if they did not have a blue forehead.

Q: Why does the number of logicians matter in determining when to leave?

The increasing number helps each logician eliminate the possibility of not having a blue forehead and allows them to conclude they must have a blue forehead.

Q: What would happen if there were two logicians with blue foreheads?

Both logicians would initially assume they don't have a blue forehead. However, after observing that the other logician didn't leave, they both conclude they have blue foreheads and leave together.

Q: Does this logic hold true for any number of logicians?

Yes, the logic applies to any number of logicians. If there are n logicians with blue foreheads, it takes n rounds of observation for them to simultaneously leave.

Summary & Key Takeaways

• A group of 100 logicians with blue foreheads enter a room and are told at least one person has a blue forehead.

• Each time the lights are turned on, if a logician sees 99 others with blue foreheads, they don't leave the room.

• After the 100th time, all logicians with blue foreheads leave simultaneously.