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Stumped on a proof/logic problem

This strictly is more of a logic puzzle rather than a maths problem, but I did find it in a problem set for a "mathematics for computer science" course (essentially a course on discrete maths) so I'll put it here. The question is as follows,

Albert announces to his class that he plans to surprise them with a quiz sometime next week.

His students first wonder if the quiz could be on Friday of next week. They reason that it can’t: if Albert didn’t give the quiz before Friday, then by midnight Thursday, they would know the quiz had to be on Friday, and so the quiz wouldn’t be a surprise any more.

Next the students wonder whether Albert could give the surprise quiz Thursday. They observe that if the quiz wasn’t given before Thursday, it would have to be given on the Thursday, since they already know it can’t be given on Friday. But having figured that out, it wouldn’t be a surprise if the quiz was on Thursday either.

Similarly, the students reason that the quiz can’t be on Wednesday, Tuesday, or Monday. Namely, it’s impossible for Albert to give a surprise quiz next week. All the students now relax, having concluded that Albert must have been bluffing. And since no one expects the quiz, that’s why, when Albert gives it on Tuesday next week, it really is a surprise!

What do you think is wrong with the students’ reasoning?

The argument that the test cannot be on Friday seems fine to me, since it really will not be a surprise by the end of Thursday if there hadn't been a test for the entire week. The other arguments for the other days also seems logically correct, at least to prove that the test won't be on the respective days.

I'm suspecting it has to do with the way the cycle leads to the conclusion that the test cannot occur at all, rather than the individual arguments to show that if the test isn't on day X, it isn't on the previous day either. However I'm not seeing any way to show this, I'm not seeing exactly where the logical fallacy is either.

Can anyone shed some light on this problem please? The more I think about it, the more and more confused I get. :confused:

Reply 1

donutellme

The flaw is in their second part. If the test hasn't been given by Wednesday, then it could be given on Thursday OR Friday. However, the students assume that their reasoning for it not being on Friday still stands, which it doesn't since Thursday hasn't arrived yet.

Reply 2

ctrls

Original post by donutellme

That makes sense, but I'm not fully seeing why it can be on Friday. If Albert sets it on Friday then it will no longer be a surprise, so would it be wrong to conclude that it is impossible for him to set it on Friday?

Reply 3

donutellme

Original post by ctrls

That makes sense, but I'm not fully seeing why it can be on Friday. If Albert sets it on Friday then it will no longer be a surprise, so would it be wrong that it is impossible for him to set it on Friday?

Perhaps. Because if it hasn't happened after Thursday then the students will know that it's on Friday, which will, in a sense, take away the surprise. However, if it is hasn't happened after Wednesday, then it could be either Thursday or Friday, so there is still a surprise element left. :smile:

So yes, I guess it's impossible for it to be on Friday. But only Friday.

Reply 4

funkymonkey_007

The logic for Thursday (or perhaps it first arises at Wednesday?) doesn't follow because the conditions change.

Firstly, as you said the test can't be on Friday because, IF IT HADN'T OCCURRED BY FRIDAY (i.e. it didnt occur on Monday, Tuesday, Wednesday or THURSDAY), it would no longer be a surprise since then it must occur on Friday which isn't much of a surprise.

Note that in the question, because of the chronology, the assumption that it can't occur on Friday is based on the fact that it already hasn't occurred in the week.

Then, to prove that it can't be on Thursday, they assume that it won't occur on Friday...but this assumption is in turn based on the fact that it already hasn't occurred on Thursday.
So once you try to find if the test will occur on Thursday, the inferences made based on the assumption that it won't occur on Thursday are invalid.

In other words, they prove that it can't occur on Thursday based on the assumption that it already hasn't taken place on Thursday.

Hope this helps.

(edited 9 years ago)

Reply 5

ctrls

Original post by funkymonkey_007

The logic for Thursday doesn't follow because the conditions change.

Firstly, as you said the test can't be on Friday because, IF IT HADN'T OCCURED BY FRIDAY (i.e. it didnt occur on M,T,W,T), it would no longer be a surprise since then it must occur on Friday which isn't a surprise.

(Note that because of the chronology, the assumption that it can't occur on Friday is based on the fact that it already hasn't occured in the week)

Then, to prove that it can't be on Thursday, they assume that it won't occur on Friday...but this assumption is in turn based on the fact that it already hasn't occured on Thursday.
So once you try to find if the test will occur on Thursday, the inferences made based on the assumption that it won't occur on Thursday are invalid.

In other words, they prove that it can't occur on Thursday based on the assumption that it already hasn't taken place on Thursday.

Hope this helps.

I can see what you are saying and you're probably right, but I'm still struggling to convince myself that it is. I think I've almost got it though, I'll think over it a bit more and see if I can figure it out.

Reply 6

funkymonkey_007

Original post by ctrls

I'm not entirely sure either :P I just threw it out there as a thought or 2 but I think it is something along those lines anyway... Was probably all over the place cause I was 'thinking out loud' if you will :P

It is a nice thought question anyway :biggrin: