Paradox of the Unexpected Exam

Date: 03/26/98 at 19:52:20
From: Jennfier McDaniel
Subject: Paradox problem

The paradox of the unexpected exam... 

A teacher announces that a test will be given on one of the five 
weekdays of next week, but tells the class, "You will not know which 
day it is until you are informed at 8 a.m. of your 1 p.m. test that 
day." Why isn't the test ever going to be given?

I would greatly appreciate your help solving this problem. I can't 
seem to make any sense of it.  

Thank you,

Jennifer McDaniel

Date: 03/26/98 at 22:10:31
From: Doctor Sam
Subject: Re: Paradox problem

Jennifer,

This is a variation on a paradox called "The Unexpected Hanging," 
which you can read about in a book by Martin Gardner by the same 
title. The logic of the "unexpected exam" is this:

The exam cannot be given on Friday because, if it were, then Thursday 
evening you would KNOW that the exam had to be on Friday, since that 
is the only day left.

But the exam cannot be given on Thursday as well. If it is Wednesday 
night and you have not yet been given the exam then you know that it 
must be given either Thursday or Friday. But we have already decided 
that it cannot be given on Friday... so it MUST be given on Thursday. 
Having reasoned that the exam will take place on Thursday, you will no 
longer be surprised to see the exam on your desk at 1:00 p.m. So the 
professor CANNOT give the exam on Thursday.

You can back up through the week in similar fashion. Since it cannot 
be on Thursday or Friday, then by Tuesday evening you would KNOW that 
it had to be on Wednesday and, therefore, you wouldn't be surprised... 
and so it CANNOT be on Wednesday. Or on Tuesday. Or on Monday.

Therefore it cannot be given.

Of course, when you walk into class Tuesday and find that it is exam 
day you will be surprised! That is the paradox.

-Doctor Sam,  The Math Forum
 http://mathforum.org/dr.math/   

Date: 06/20/2000 at 22:56:48
From: Ian Wang
Subject: Answer to problem

I was looking at the "Paradox of the Unexpected Exam" and I believe 
that I have found the flaw in the reasoning, but correct me if I'm 
wrong.

The first part of the proof uses the condition "if the test has not 
taken place by Thursday night, then..." in order to prove that the 
test cannot take place on Friday.

The next part of the proof then attempts to prove, using the same 
method, that it cannot be on Thursday. However, here the error occurs. 
You need to know that it's not on Friday to know that it can't be on 
Thursday, but if you know it's not on Friday, then by the given, it's 
already past Thursday. Thus this is not a paradox, just a cleverly 
phrased teaser.

Ian

Date: 06/22/2000 at 13:19:26
From: Doctor Twe
Subject: Re: Answer to problem

Hi - thanks for writing to Dr. Math.

You ask a good question. There's a subtle distinction between when 
we're doing the reasoning and when we know for sure. If I am told on 
Monday that on Friday, $100 will be deposited in my checking account, 
I can make plans to pay my bills on Saturday - even though it's not 
Friday yet, and the money isn't in my checking account yet.

Similarly, on the weekend preceding the week of the exam, the student 
can reason about the upcoming week. He or she can deduce that it won't 
be given on Friday (for the reason cited). Thus, even though it is not 
yet Thursday afternoon, we know that it can't be given on Friday - if 
it were, a contradiction would result. The contradiction wouldn't 
occur until Thursday afternoon - but we know that it would indeed 
occur (if the exam were given on Friday). Thus we know that it must be 
given between Monday and Thursday. But we've figured this out before 
Monday has even arrived.

So Friday is out, and we can think of it as a four-day week. Note 
that we're still doing all this reasoning the preceding weekend. Now 
we can eliminate Thursday using similar logic. The contradiction 
wouldn't occur until Wednesday afternoon - but it still must occur. We 
can proceed to eliminate every day in that manner.

As a counterexample, suppose the exam is given on Thursday. Let's 
speed the clock to Wednesday night. We then think, "The test could be 
tomorrow or Friday. But if it is on Friday, I'll know that tomorrow 
afternoon - before the 8:00 a.m. announcement on Friday. So the teacher 
can't give it to me on Friday and still have it be a surprise. So it 
must be tomorrow. But since I now already know that it'll be given 
tomorrow, that isn't a surprise either."

We can extend this example (with more "but if..."s) for Wednesday, 
Tuesday, and Monday as well. No matter what day the teacher chooses to 
give the test, the student will know the night before by eliminating 
the days that follow.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/