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Paradox Of The Unexpected Egg

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

Note by Venture Hi
2 years, 2 months ago

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When you said that the egg can't be in box 9 then there was some logical error because assurity of the egg being in box 10 arises only when you've once opened the nine boxes, so when you have opened eight boxes then also the appearance of egg in box 9 would be unexpected as it could have possibly been in both box 9 as well as box 10. Samagra Sharma · 2 years, 2 months ago

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@Samagra Sharma I don't see how this is a paradox. I agree with you, Samagra. When you reach box 9, you're not sure whether it's in box 10 or box 9. Hence, in this case, an egg in box 9 would be an unexpected egg. You can only rule out box 10 if you have opened box 9. If anyone sees where I'm going wrong, please tell me! Ryan Tamburrino · 2 years, 2 months ago

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@Ryan Tamburrino Yes Ryan, this seems to be a correct explanation. A more evolved form of this paradox is The Unexpected hanging paradox (see wikipedia) Samagra Sharma · 2 years, 2 months ago

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@Samagra Sharma Actually because your friend has told you that you will at no point be able to know an egg is inside a box until you actually open the box you can rule out future scenarios. That statment tells you that at no point in the future will you be able to deduce an egg is in a given box. Therefore you don't have to open box nine in order to have deduced the egg is in box 10 because you can rule this out as a future scenario in which the egg would be expected which would go against the words of your trustworthy friend. This is what makes this either an impossible statment or a true paradox, whichever you prefer to call it. Bartosz Blachut · 1 year, 4 months ago

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@Samagra Sharma Let us assume that the egg is in box9 . I have opened 8boxes . For sure egg is not in box 10 . So if the condition is true then egg has to be in box9 . And I deduced it without opening . AND deduction of egg not being in box10 is also perfect . But as Siddarth said the statement made by our friend is not mathematical and thus cannot be treated like one as we have done in explanation of paradox Devansh Shringi · 2 years, 2 months ago

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@Devansh Shringi Sure, IF you ASSUME it's in box 9 I'm sure it's easy to deduct it's in box 9. But you can not assume it is in either box9 or 10 until you open box9. This is a mathematical problem known as P vs NP and can probably never be solved because in this instance the criteria can not always be met Hans Reygaert · 1 year, 4 months ago

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@Samagra Sharma Actually, it still is a paradox. But the paradox only occurs when you have removed all other variables. You can count in any direction, the last box will never be unexpected.

Your logic still is sound, but the statement that there is a truly unexpected egg is wrong. It should add if there are other boxes Hans Reygaert · 1 year, 4 months ago

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No matter what you are expecting to find an egg in one of the 10 boxes. And if the individual who put the egg in the boxes wanted to use box 10 then there is no reason why he could not. John Fairley · 1 year, 4 months ago

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Schrodingers cat..? Untill the box is opened then the egg can be both in or not in the box...so until opened and observed the event is unexpected. ?maybe?...lol. John Dowe · 1 year, 4 months ago

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Want to bet that the test will be on Saturday? Cynthia Chan · 1 year, 4 months ago

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The paradox exists here because our friend can not guarantee that there will be an unexpected egg (damn him!) and the fact that "there will be an unexpected egg" is not a statement, that is, it is neither always true, nor always false, but can be true or false, much like "Today is Sunday"

In the story, we correctly conclude that what our friend said is false. But we incorrectly assume that since it false, "there will be an unexpected egg" is also false statement.

For example, if a salesman sells you a product and guarantees that his products are not defective, but you know that he sold someone else a defective product, you can only say that his guarantee was false, not that the product he sold you is defective. It may or may not be defective.

Likewise in the story, only the guarantee is false, but that does not mean there isn't an unexpected egg present. We incorrectly assume that that there is no egg and thus don't expect any egg. We could have always expected an egg in the next box, and the egg when found would've been expected, resolving the paradox. Siddhartha Srivastava · 2 years, 2 months ago

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While going along with the problem, the person B forgot that all of person A's statements were all "absolutely trustworthy." With that said the so-called 'unexpected' egg shouldn't have been ruled out from all the boxes, even with the box-by-box deduction. I am pretty sure in logic all statements that are given should be used. Just saying. Viktor Bhattacharjee · 1 year, 2 months ago

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You won't have test on Thursday if it didn't took place on Wednesday Himanshu Goyal · 1 year, 3 months ago

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Eliminating each event in this type of paradox or each box or each day in the above paradox is totaly independent we can't mix it with each other Aman Pathak · 1 year, 4 months ago

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The professor is correct, because the student is only given a theory, that can be brocken like in the egg paradox. The Lego Man The Lego Movie · 1 year, 4 months ago

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U can comment on box 10 only when u hve seen d other boxes and u cannot reject boxes like dis because when d next time u start opening d boxes u hve to take all d 10 boxes Akarsh Kumar Srit · 1 year, 4 months ago

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If your friend places the box in box 9, why wouldn't it be considered unexpected? After reaching box 8 it would be unexpected whether the egg will be in box 9 or 10. So with all the no.s except for when the egg is in box 10. Silver Vice · 1 year, 4 months ago

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The egg is not a chicken egg. It's like, a dinosaur egg, or something. You weren't expecting that, were you? Pretty unexpected no matter which box it's in! Haha! Andrew McLaren · 1 year, 4 months ago

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How can you rule out box 10 if i was the individual hiding the egg i would hide it in the one place you are not expecting to find it. Box 10. John Fairley · 1 year, 4 months ago

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@John Fairley By the time you reach box 9, you will know that the egg is in box 10, meaning you will expect the egg. Robert Lytle · 1 year, 4 months ago

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@Robert Lytle By the time you reach box 9, in box 10 could be an unexpected chicken. John Fairley · 1 year, 4 months ago

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In response to the teacher and his class, I see where the student's logic is going, but these rules cannot be stacked on top of each other like that. If it is Thursday at the end of the class, obviously the test has to be Friday so it can be ruled out. But if it is only Wednesday, the test can still be on either Thursday or Friday. Friday cannot stay ruled out completely because the test could be given on the next day, Thursday, and it would be a surprise. The test could come any day Monday through Thursday. Nelson Earle · 1 year, 4 months ago

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There was a time when people thought of sandwiching Sunday between Wednesday and Thursday, but then the idea was dropped as it failed to divide the work week. Rajen Kapur · 1 year, 4 months ago

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The problem is each time you open a box, and there's no egg, then there's a greater chance of there being an egg.

Before opening anything there is a 1/10 chance it's in the first box.

Then after you open one box there's a 1/9 chance. Then 1/8, 1/7, 1/6... all the way till you're left with the last box where the chance is 1/1. So if it's in the last box it will be expected.

The error in logic responsible for this paradox occurs when you remove the 10th box from the set as a possible candidate for the egg when calculating the probabilities. This will result in all of the probabilities changing, and repeating this error eventually results in a 1/1 chance for each box, and causing the box to be removed as a potential candidate. Matt Mcc · 1 year, 4 months ago

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It could be the fact that you reasoned this way that made the egg unexpected. P L · 1 year, 4 months ago

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This paradox references itself and contradicts itself at the same time. Knowing the second law of thermodynamics to be true, the problem ultimately finds a solution right in zee middle.Nature abhors a vacuum :) Andrew Mackintosh · 1 year, 4 months ago

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Since the egg can only be in one box, you can only rule out a maximum of a single box per turn. Saying "It cannot be in box 9 and box 10," is wrong because it would be the same as saying it cannot be in any of the 10 boxes when in fact you wouldnt be able to make a logical guess as to a single box for the egg to reside in, thus making it a truly unexpected location. Andrew Maksymiuk · 1 year, 4 months ago

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You cannot aplly the BOX 10 logic to BOX 5. It sounds pretty reasonable until BOX 8, but it cannot be expected after that, since that friend is not so foolish, logically. Vikram Venkat · 2 years, 2 months ago

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I think the logic behind the fact that the egg cannot be in box 10 is sound because that would guarantee an expected egg. But discounting the boxes after box 10 as possibilities is where the logic fails. The logic only comes into play at the point where all the boxes checked so far are empty. To put it another way, if your logic is so sound that there can not be an egg in a particular box, the fact that the egg is in there makes it unexpected. Lol. I'm sure I'm wrong and there's a more sensible answer to this Olawale Olayemi · 2 years, 2 months ago

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@Olawale Olayemi Another similar paradox can be found in the story written by Robert Louis Stevenson titled The Bottle Imp. Venture Hi · 2 years, 2 months ago

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@Olawale Olayemi Good explanation which is why this is a paradox. Venture Hi · 2 years, 2 months ago

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[I think] It is written here that we need to open the boxes in serial order. But, we opened it from the end. Hence our assumption is wrong. Rajdeep Bharati · 1 year, 4 months ago

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In a nutshell, the logic is faulty as it would only be applicable when someone goes into the future at a time where he has opened the nine boxes which were empty. Then and only then one can say that the egg will be inside the 10th box.As the result is certain now, the person then will go to a time where he has opened 8 boxes which resulted being empty. Then and only then that person can strike the 9th box out. Similarly he will eventually strike out all the boxes. Therefore this logic is only possible when we will be able to manipulate time ;) Yatharth Chowdhury · 1 year, 4 months ago

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@Yatharth Chowdhury That's interesting. It can only be a supprise quiz if he said nothing about it at all. Otherwise you are already expecting it over the next week and it will not be a surprise whatever day he choses! Kevin Sturge · 1 year, 4 months ago

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When you deduced that the egg could not be in any of them, you no longer expected there to be an egg. That means it was an unexpected egg. Ruby Syuukyoku · 1 year, 4 months ago

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This is identical to the situation of a teacher announcing that there will be a surprise test the next week, on any day from Monday to Friday. Raj Magesh · 2 years, 2 months ago

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