There are many versions of this puzzle including the original version in which an air raid warden announces that there will be a surprise drill at 8 p.m. some night in the next seven days. Of course, such a statement could be said of a pop quiz or any military drill. Part of the controversy in each case is the question as to whether this sort of statement can be absolutely true.
Gardner does a very good job of outlining the paradoxical nature of the problem. On the one hand, the prisoner's lawyer argues that if the prisoner is not hanged by Friday, then the predicted surprise can not occur on Saturday at noon since then the judge's statement would have been false. Hence the purported contradiction to the judge's pronouncement of sentence. But we must assume that the judge's sentence is a statement of fact and so the prisoner must be hanged on some day before Saturday. If this must be true, then we can conclude that if the prisoner is not hanged by Thursday, then he will know that the judge can not delay the hanging until Saturday and so will know for a certainty that he can not be hanged on Friday.
Likewise, if the prisoner is not hanged on Wednesday, he will know that he cannot be hanged on Friday or Saturday and so he must expect the hangman to come on Thursday. But the hangman can not come when expected, so He can not be hanged on Thursday, Friday or Saturday.
The argument continues in this fashion so that it is clear to the prisoner that there is no day which the judge may pick for which the prisoner will not expect the hangmen. Secure in this knowledge the prisoner does not expect the hangman when he appears on Wednesday at noon. See comments for a discussion
See the Mathematics Resolution link on this blog for a more detailed analysis.
ReplyDeleteOne of the problems with conundrums of this type is that many of the words and phases in the description of the problem are ambiguous. One sometimes has the feeling that this is an intentional tease. When one attempts to give a mathematical interpretation, a great deal of the of the ambiguity disappears but there is also the tendency to narrow the confines of the original problem which is not always satisfactory to the reader.
Let me just mention here that the problem can be simplified to 7 boxes labeled 1 to 7 with a egg hidden inside one of the boxes. Someone is asked to open the boxes and that they will be surprised to find an egg in one box.
An additional point to be made is that the Lawyer's argument is not sound because it begins with the premise that the prisoner is alive on Friday instead of admitting that it were possible that the prisoner is dead and was surprised.