The surprise test
A teacher promises a surprise test next week. The class proves it can't happen.
Also called the unexpected hanging; first published by D. J. O'Connor, 1948
Mrs Kulkarni's promise
On a Friday afternoon in Pune, Mrs Kulkarni told Class 8B: "Next week, Monday to Friday, there will be one maths test. And it will be a surprise: on the morning of the test you will not know it is that day."
Aditya, the class's chief arguer, got to work. "It can't be Friday," he said. "If there's been no test by Thursday afternoon, we'd know it's Friday, so it wouldn't be a surprise." The class nodded.
"So the last possible day is Thursday. But then, if there's no test by Wednesday afternoon, we'd know it's Thursday. So not Thursday either." Wednesday went the same way. Then Tuesday. Then Monday. "There can't be a surprise test at all!" Aditya declared, and nobody bothered to revise.
On Wednesday morning Mrs Kulkarni handed out the test papers. Everyone was completely surprised.
The puzzle
The students seem to prove that a surprise test is impossible, and yet the test is a surprise.
See it for yourself
What's really going on
The students' chain of reasoning assumed, at every step, that the teacher's announcement was true and that they knew it was true. But their conclusion was that the announcement was false. Once you've decided the teacher's promise can't be trusted, you can no longer use it to rule out any day, so Wednesday is a genuine surprise.
The paradox is really about knowledge: what you can know about your own future knowledge. Philosophers and logicians have written hundreds of papers on it, and they still disagree about exactly which step of Aditya's argument is the faulty one.
Still argued about. Everyone agrees the test can be a surprise; there is less agreement on precisely where the students' logic breaks.
Think about it
Does the argument still work if the test could be on any day in the next year? What if there are only two possible days?