An Unexpected Hanging

I came across a very intriguing paradox in a Martin Gardner book, titled An Unexpected Hanging And Other Mathemaical Diversions. I was very interested in how deeply the paradox divides logic from reality and in how many ways it can be portrayed. It appeared in the 1940’s and was discussed by Professor Michael Scriven in 1951.

The Story

The story of the paradox goes like this. A prisoner is sentenced by a judge to be hanged on Saturday. The judge says to the prisoner “The hanging will take place at noon, on one of the 7 days of next week. But, you will not know which day it is until you are informed on the morning of the hanging”. The judge is an honest and fair man (let us assume he does not lie) and as soon as the prisoner and his lawyer were alone, the lawyer broke into a grin. The lawyer stated that the judge’s decree could never be carried out in practice. How did the lawyer reach this conclusion?

The Lawyer’s Solution

The lawyer’s explanation went like this. If the judge were to hang the prisoner on next Saturday(the last of the 7 days the judge can hang him on), the prisoner would already know on Friday afternoon, since the judge could only hang him on Saturday by then. Likewise, Friday could be ruled out, since by Thursday, the Judge could only hang him on Friday or Saturday, and since he cannot hang the prisoner on Saturday, he can only do so on Friday. The Judge’s decree would be violated. For every day of the week, the hanging can be ruled out be this logic.

Weirdly enough, the judge’s statement does not appear to be able to be carried out unless he is a liar, which he isn’t. But, the prisoner is hanged on Thursday, completely unexpected!

Explaining the Paradox

There are a couple reasons why this paradox has occured and has befuddled the prisoner. The fact of the matter is that the prisoner will be hanged and although the lawyer’s logic is sound, it completely forgets this fact.

One way I devised to demonstrate this is a slight variation of an example shown in Martin Gardner’s book. Suppose you and your friend are both perfect logicians. There are ten cards, Ace to Ten on the table, but not in that order. Your friend will start to flip each card and you must prove where the Ace is. This means, using iron hard logic to confidently say where the Ace is before your friend flips over the card. Your friend offers you a 1000 dollars if you deduce where the Ace is.

Now, using the same logic as in the story, you know your friend would not have put the Ace in the tenth spot. After flipping the first 9 cards and not seeing the Ace, you would know the Ace was in the tenth spot and you would have won. You similarly eliminate the other 9 spots and confidently say that the Ace is not even in the ten cards. Assuming your friend is truthful and wants to win the bet, have you won the bet?

Conclusion

Sadly you have not won the bet. The logic you used to justify your claim is perfectly sound but neglects a single key point that is there in the story and example: The prisoner must by hanged and the Ace is in the cards. Overlooking this key fact enables you to think you have won, since your logic is iron hard. But, the reality is that your friend has probably won the bet(if they have not put it in the tenth spot). This paradox is extemely interesting to and I hope it is to you as well.