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This version of well-known puzzle appeared in the 'Skeptical Adversaria' (the newsletter ASKE, the Association for Skeptical Enquiry), September 2004.
This little problem continues the theme of 'intuition vs. logic' (by no means a clear-cut distinction, but a useful one nonetheless). As with the puzzle in the previous issue, intuition immediately informs us of what the correct response is, but after the application of logic (and, if necessary, mathematics, which some would say is a branch of logic) another answer emerges. But this puzzle has a twist in the tail. Intuition has its say, but is superseded by logic; yet intuition continues to protest – surely the original 'gut' reaction is the correct one and on this occasion, logic must be at fault?
Here is the puzzle. The scene is a courtroom in the days when capital punishment was meted out in this country. After a relatively short period of retirement, the jury returns and the foreman delivers the verdict on the prisoner, who has been accused of a most dreadful murder. 'We find the prisoner "Guilty" my lord.'
The only sentence that the learned judge is empowered to delivery is death by hanging. But is there something in the wise old judge's demeanour that signifies his disquiet that the jury is willing to convict a man on the basis of such slender evidence? And is this the reason why he sets such a deceptively complex condition on the death sentence?
To the consternation of the court, the judge sends for the hangman to be present. He then turns to the defendant.
'Prisoner at the bar, you have been found guilty of a most foul murder and I sentence that you will be taken from this place to be hanged by the neck until you are dead'.
He then turns to the hangman and says, 'Sir, I charge that you hang this man at noon on Monday, Tuesday or Wednesday of next week'. Then, sternly pointing his finger at the hangman, he continues, 'Under no circumstances must the prisoner know in advance on which day his hanging is to be until you are ready to send for him. You must now choose in your own mind the day of the hanging. If you cannot do this, I shall order the prisoner's release'.
At first the hangman looks bemused and appears to be on the point of announcing that he has come to his decision when he suddenly stops and begins stroking his chin in a thoughtful manner. Once again he appears to be ready to announce that he has made his choice, but suddenly once more he stops and his face gradually assumes a worried and confused expression. His growing discomfort is apparent for all to see and eventually, with a shrug of his shoulders and in an apologetic and deferential manner says, 'My Lord, you have set me an impossible task'.
With a barely suppressed air of triumph, the judge turns to the prisoner and says, 'Sir, you leave this court a free man'.
As the courtroom clears, the hangman is seen, on the point of leaving, to turn back as if to make a belated announcement that he has finally made up his mind. But then he stops, shakes his bowed head and slowly carries on his way.
Why is the hangman unable to choose the day? Is he correct not to do so?
The answer is given below, so do not look until you are ready.
At first it seems that the hangman is able to fulfil the judge's order quite simply by choosing one of the three days at random. The prisoner certainly would not know which day was thus selected. But then the hangman realises that the three days (Monday, Tuesday, Wednesday) are in sequence. This means that if he chooses Wednesday as the day of the hanging, then as soon as noon on Tuesday has passed the prisoner will know that the day will definitively be Wednesday. Thus the judge's stern instructions will be disobeyed. So Wednesday is out. No problem: the hangman can make a random choice between Monday and Tuesday. The prisoner will still be unable to predict which one is drawn. But then the hangman thinks, 'The prisoner may well realise that I cannot choose Wednesday, leaving only Monday and Tuesday. And then he will realise that I cannot select Tuesday because when noon is passed on Monday, he will know that he will be hanged on Tuesday. So Tuesday is out. So the prisoner will realise that I can only choose Monday. Therefore I cannot carry out the judge's instructions'.
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