Surprise exam paradox
Categories: recreational maths paradox
The surprise exam paradox is a paradox that relates to our knowledge of future events. It is also known as the unexpected hanging paradox if you prefer a more macabre scenario.
It was first popularised by the maths and science writer Martin Gardener in the 1960s, in his excellent Mathematical Games column in Scientific American magazine. The paradox was originally proposed by mathematician Lennart Ekbom in the 1940s.
To date, there is no generally accepted resolution to the paradox. We will look at a few of the popular attempts at solutions here.
The paradox
A maths teacher tells her class that she will be setting an exam in the next week, but that the exam will be a surprise to them. They won't know what day the exam will be until they arrive in the class and she hands out the paper.
The class, who are all grade A students, discuss the situation and conclude that there will be no exam.
Their reasoning is as follows. The exam couldn't be on Friday, because on Thursday evening, if they hadn't already had the exam, they would know the exam was going to happen on Friday. That would contradict the teacher's assertion that the exam would be a surprise.
But the exam couldn't be on the Thursday either. On Wednesday evening, if they hadn't already had the exam, and knowing that the exam couldn't be on the Friday, they would know that the exam must be on the Thursday. That again would contradict the teacher's assertion that the exam would be a surprise.
By the same logic, they concluded that the exam couldn't be on Wednesday, or Tuesday, or Monday. So, secure in the knowledge that the exam wasn't going to happen, they didn't bother to revise.
When the students turned up to their class on Wednesday, the teacher handed out the exam papers - to the complete surprise of everyone in the class.
A paradox within a paradox
The interesting thing about this paradox is that it has a double payload.
The first paradox is that the teacher appears to have made a statement that contradicts itself, and so is impossible.
The second paradox is that she does in fact manage to deliver on both of her contradicting promises! This aspect makes the paradox more interesting than, for example, the liar's paradox mentioned below.
Solving the problem logically - first attempt
We can attempt to apply formal logic to the problem. The main problem with this approach is that the term "surprised" has no well-defined meaning.
We could define surprise to mean non-deducible, like this: The exam will take place next week and the day of the exam will not be deducible the night before from the assumption that the exam will take place next week.
According to this, if it gets to Thursday evening and the exam has not already happened, then we can deduce that it will happen on Friday.
But on Wednesday evening, all we know is that the exam hasn't happened yet. Based only on the fact that we know it must happen that week, we can deduce that it must happen on either Thursday or Friday.
So provided the teacher doesn't leave the exam until Friday, there is no paradox.
Solving the problem logically - second attempt
But hang on, haven't we missed something here? Our definition of surprise deliberately excluded the fact that the teacher had told her class they would be surprised. If we ignore that key fact, then of course the problem is easy to solve, because we have avoided the very thing that makes it difficult.
So let's try a different definition: The exam will take place next week and the day of the exam will not be deducible the night before from the teacher's statement.
This exposes a flaw in the formulation of the problem. Our definition of "surprised" relies on the teacher's statement that they will be surprised by the exam.
This is self-referential, but more importantly, it is inconsistent. It is not dissimilar to the famous liar's paradox, where someone says "I am lying". If they are lying, then that statement is not true, so they are not lying. But if they are not lying the statement is true, so they are lying.
Ultimately, if a logical statement is inconsistent then you cannot deduce anything from it. So the class can reject everything the teacher has said about the day of the exam, so they don't know which day next week the exam will be.
Of course, this also assumes you accept the new definition of "surprised". Some people don't see the problem as a logical problem at all.
Reducing the problem to a single day
There is a school of thought that considers the final day to be the only significant part of the paradox, and what happens earlier in the week is just a distraction.
In that case, we should consider the situation if the teacher announced the exam on Thursday evening.
She would effectively be saying that the exam will be tomorrow, but they won't know what day the exam will be until they arrive in the class and she hands out the paper.
Of course, this sounds like nonsense. It is a type of Moore's paradox - the classic example of this paradox is someone making the statement "It is raining, but I don't think it is raining". This is nonsense but for a very particular reason:
- The idea that it is raining but you don't think it is raining is quite plausible. If you are indoors it might have started raining and you just didn't notice.
- But to simultaneously say that it is raining and you don't think it is raining is nonsense because you have already acknowledged that you know it is raining.
In our case, what the teacher is saying falls into the first category. She is saying that she knows that the exam will be tomorrow, but that her students will be surprised when it happens.
That is equivalent to her saying "It is raining but my students don't think it is raining". There is no contradiction in that statement.
An epistemological solution?
Some find the previous solutions problematic because the paradox appears to be more related to a question of knowledge rather than logic. In philosophy, epistemology is the study of knowledge, and in particular the question of whether a particular belief is justified or not.
If we take the single-day version of the problem, on Thursday the teacher makes two statements:
- The exam will be tomorrow.
- The students will be surprised if the exam is tomorrow.
There is an important distinction between what the students have been told, and what they know. They have been told those two facts, but they don't know that either will be true.
Given that there is a problem with assuming that both are true, the students should consider the possibility that one or both are not true. But there is no obvious reason to assume that one is more likely to be false than the other. So the students don't know whether the exam will take place on Friday or not.
If we consider the original problem, where the exam is announced the week before, then there is another potential problem. How could the teacher know whether or not the students are going to be surprised or not when the exam is given on a Wednesday? Maybe she has not considered the paradox her students identified, or maybe she has considered it and rejected it for some reason (perhaps one of the reasons given in this article). Alternatively, maybe her students will be unsurprised when the exam is given on Wednesday for some other reason.
Either way, her assertion that the students will be surprised is at best an opinion not a fact.
See also
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