The barber paradox
A barber shaves everyone who doesn't shave himself. Who shaves the barber?
Bertrand Russell's set paradox, 1901; barber version from his 1918 lectures
The rule of Ramgarh
In the small town of Ramgarh there was one barber, Mr Iqbal, and one very strict rule painted on his shop: "Iqbal shaves every man in Ramgarh who does not shave himself, and only those men."
It worked perfectly. Men who shaved at home never came in. Men who didn't, came to Iqbal. Everyone in town was either a self-shaver or an Iqbal customer.
Then one morning a schoolgirl named Fatima stopped outside and asked: "Mr Iqbal, who shaves you?"
If Iqbal shaves himself, he is a man who shaves himself, so the rule says he must not shave him. If he does not shave himself, he is exactly the kind of man the rule says he must shave. Iqbal put down his razor and sat very still.
The puzzle
Every possible answer to "Does the barber shave himself?" breaks the barber's own rule.
See it for yourself
What's really going on
The way out is surprisingly simple: no such barber can exist. The rule describes something impossible, the way 'a square circle' does. Fatima didn't find a weird barber; she proved the sign was impossible to obey.
The barber is a friendly version of a real crisis in maths. In 1901 Bertrand Russell found the same trap in set theory: the set of all sets that do not contain themselves. Does it contain itself? Mathematicians fixed this by rebuilding set theory with careful rules about which sets are allowed to exist, the system most maths uses today.
Think about it
Can you write a rule for a school club that it would be impossible for the club secretary to follow?