Imagine that you have before you ten boxes labeled from 1 to 10. While your back is turned, a friend conceals an egg in one of the boxes. You turn around. “I want you to open these boxes one at a time,” your friend tells you, “in serial order. Inside one of them I guarantee that you will find an unexpected egg. By ‘unexpected’ I mean that you will not be able to deduce which box it is in before you open the box and see it.”
Assuming that your friend is absolutely trustworthy in all statements, can this prediction be fulfilled? Apparently not.
Your friend obviously will not put the egg in box 10, because after you have found the first nine boxes empty you will be able to deduce with certainty that the egg is in the only remaining box.
This would contradict your friend’s statement. Box 10 is out.
Now consider the situation that would arise if your friend were so foolish as to put the egg in box 9. You find the first eight boxes empty. Only 9 and 10 remain. The egg cannot be in box 10.
Ergo it must be in 9. You open 9. Sure enough, there it is. Clearly it is an expected egg, and so your friend is again proved wrong. Box 9 is out.
But now you have started on your inexorable slide into unreality. Box 8 can be ruled out by precisely the same logical argument, and similarly boxes 7, 6, 5, 4, 3, 2 and 1. Confident that all ten boxes are empty, you start to open them. What have we here in box 5? A totally unexpected egg! Your friend’s prediction is fulfilled after all.
Where did your reasoning go wrong?
Moderator's Addendum: This paradox is also seen in the following form
The professor said to his class: I'll be giving you a test one day next week, but you won't know in advance which day. But that's impossible! objected one student. If we haven't had the test by end of Thursday's class, then we'll know in advance it must be on Friday. So we can't possibly have the test on Friday, or we'd know in advance.
That leaves only Monday, Tuesday, Wednesday and Thursday as possible days.
But if we haven't had the test by end of Wednesday's class, we'll know in advance it must be on Thursday, since we have already ruled Friday out. So we can't have the test on Thursday either, or we'd know in advance.
That leaves only Monday, Tuesday and Wednesday as possible days.
But if we haven't had the test by end of Tuesday's class, we'll know in advance it must be on Wednesday, since we have already > ruled Friday and Thursday out. So we can't have the test on Wednesday either, or we'd know in advance.
That leaves only Monday and Tuesday as possible days.
But if we haven't had the test by end of Monday's class, we'll know in advance it must be on Tuesday, since we have already ruled Friday and Thursday and Wednesday out. So we can't have the test on Tuesday either, or we'd know in advance.
That leaves only Monday as a possible day. So we'll know in advance that the test will be on Monday. So we can't have the test on Monday either, or we'd know in advance.
So we can't have the test at all.
Who is right, the professor or the student? I have never seen a satisfactory resolution of this paradoxical argument.
From Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging