Blue Forehead Room Riddle Solution: Why All 100 Logicians Leave at Once | Puzzles | Math for Fun and Glory | Khan Academy

TL;DR

All 100 blue-forehead logicians leave the room simultaneously right after the 100th time the lights are turned on. Each was told at least one person has a blue forehead and that everyone is a perfect logician. The trick is reasoning up from smaller cases: with 1 person he leaves immediately, with 2 they leave on the second round, and with n it takes n rounds. Read on to see how each logician deduces their own hidden color.

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: What is the solution to the blue forehead room riddle?

All 100 logicians leave the room at the same time, right after the 100th time the lights are turned on and then off. Even though nothing visibly changes between rounds, the number of rounds that pass is what lets each person deduce their own forehead is blue.

Q: How does a single logician know they have a blue forehead?

With only one person in the room, he looks around and sees no one else. Since he was told at least one person has a blue forehead and he is the only one there, he concludes it must be him and leaves as soon as the lights go off.

Q: What happens when there are two logicians with blue foreheads?

Each one sees the other's blue forehead and reasons: if I did not have a blue forehead, the other person would see no blue forehead, know theirs must be blue, and leave on the first round. When the other person does not leave, each deduces their own forehead is blue, so they both leave together on the second round.

Q: Why does nothing seem to change between rounds if everyone still leaves?

Visually nothing changes because each person keeps seeing the same 99 blue foreheads every time. What changes is the information: each round that passes without anyone leaving rules out one possibility, and after enough rounds the only consistent conclusion is that every blue-forehead logician has a blue forehead.

Q: Why does the number of logicians matter for when they leave?

The count sets how many rounds of no one leaving must pass before everyone can conclude. If there are n logicians with blue foreheads, it takes exactly n rounds of observation before they all leave simultaneously.

Q: What are the logicians told before entering the room?

They are told that at least one of the people in the room has a blue forehead, and that everyone in the room is a perfect logician with infallible logic. Some even have the statement written on a card in case they forget.

Q: When are the logicians supposed to leave the room?

Once a logician has determined that their own forehead is blue, they leave the room the next time the lights are turned off. After that, the lights are turned back on, people look again, and the process repeats until everyone has left.

Q: Why are the foreheads painted blue rather than another body part?

The forehead is used because a person cannot see their own forehead, even though everyone else can see it clearly. This is what makes the puzzle work: each logician must deduce their own hidden color purely from others' behavior.

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.