Did you solve it? That Sally Rooney hat puzzle

Earlier today I set you the following puzzle, which I read in the new Sally Rooney novel, Intermezzo. Here it is again with the solution.

A liar who always lies says “All my hats are green.”

Can we conclude that he has some hats?

I extended the puzzle by making it multiple choice. Which, if any, of the following statements can we conclude from what the liar has said?

A) The liar has at least one hat.
B) The liar has only one green hat.
C) The liar has no hats.
D) The liar has at least one green hat.
E) The liar has no green hats.

Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements.

Solution A

If the liar says “All my hats are green” then the statement “All my hats are green” is false. In other words, it is not the case that all of the liar’s hats are green.

We cannot conclude B because it is possible that the liar has, say, one red and two green hats. (It may be the case that the liar has a single green hat, but we cannot conclude it with 100 per cent certainty.)

We cannot conclude D because it is possible the liar has a single red hat and no green hats.

We cannot conclude E because it is possible that the liar has a green and a red hat.

So, either A is true or C is true – either the liar has some hats or he has no hats.

And here is where we get to the crux of the problem, the notion of vacuous truth.

Imagine I have a shelf with no books on it. The statement ‘I have read all the books on my shelf’ is technically true, although it is meaningless because there are no books. Likewise, the statement ‘I have read none of the books on my shelf’ is also technically true and meaningless for the same reason. A logician would call both of these statements vacuously true. They are true, but say nothing.

Now back to the liar. Imagine he has no hats. If he has no hats, then the statement “All my hats are green” is true (and vacuous). Which contradicts the fact that he only says false statements.

The liar must have some hats, thus A is the right answer.

I hope you enjoyed today’s puzzle. (I do recommend the Rooney book BTW.)

I’ll be back in two weeks.

The original source of this problem (in extended form) seems to be a Brazilian maths exam from 2022.

Looking for a Christmas gift for the puzzle lover in your life? My latest book is Think Twice: Solve the Puzzles That (Almost) Everyone Gets Wrong, a collection of counter-intuitive conundrums that make you think about thinking – while enjoying the pleasure of being misled.

Think Twice: Solve the Simple Puzzles (Almost) Everyone Gets Wrong. To support the Guardian and Observer, order your copy at guardianbookshop.com. Delivery charges may apply. (In the US, the book is called Puzzle Me Twice.)

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.