# Blue forehead room solution | Puzzles | Math for fun and glory | Khan Academy | Summary and Q&A

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April 14, 2009
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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.

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### 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.