The Paradox of the Condemned Prisoner

The prison governor goes to visit a condemned prisoner in his cell. He tells him that he is due to be executed at midday one day in the following month, but he won’t know in advance which day it is.

Actually, condemned prisoners are a bit of a gloomy subject for a maths blog: let’s change the scene to a school and start again!

The head teacher of a school announces that there will be a fire drill at midday one day the following week, but that students won’t know in advance which day it is. A bright (and probably annoying) philosophy student has a think about this, and then tells his friends that there can’t possibly be a fire drill. This was his argument:

“The fire drill can’t be scheduled for Friday because, once it gets past midday on Thursday, we’ll know it has to be Friday. So that would mean we would know in advance.”

“But it then follows that the fire drill can’t be on Thursday either, because once midday has passed on Wednesday – knowing it won’t be on Friday – we would then know it has to be Thursday. Once again, we would know in advance.”

“Now work back to the beginning of the week and, using the same argument, the fire drill can’t be held at all!”

He was very persuasive and, although they all felt there must be something wrong with the argument, his friends ended up agreeing with him. And then, much to their surprise, the fire drill was held on Tuesday – and no-one had known about it in advance (except, presumably, the head teacher).

So, is there anything wrong with the logic? The answer is “no”. You could argue about the semantics: what exactly is meant about “knowing in advance”, for example. But the whole situation can be expressed as a precise logical argument, and everything checks out. As the title of the blog suggest, we have a genuine paradox here. Perhaps we could argue that the head teacher’s statement is an impossibility – that what she has said will happen can’t possibly happen. It reminds me of another lovely paradox: take a card, and on one side of it write: “The statement on the other side of this card is true.” And then on the other side, write: “The statement on the other side of this card is false.” If you work through this, you’ll find that it’s impossible for both statement to be true at the same time – but that doesn’t stop us writing them down!

This simple paradox has led to much discussion, debate and learned papers over the years. A web search on “Unexpected Hanging Paradox” led to 842,000 results. So, for Maths Studies students who find the logic section of the syllabus a bit tricky, just relax: things could be a lot worse!