Hard maths problem for you (Warning: VERY hard!)

[TheCenturion]

This is genuinely hard, I would be very surprised and imrpressed by a member who could get the correct answer and doesn't have any maths qualifications above A level!

I can't quite remember it correctly, I think that this is how it goes. If I write it wrong, I'll edit it and tell you

A mysterious stranger (paedo bear maybe?) talks to Sam and Paul. He thinks of two integers between 2 and 99 inclusive. He tells Paul the product of the two numbers, and tells Sam the sum of the two numbers. Paul and Sam then talk to each other, without saying the number that they've been told. They do however, know the nature of the number that the other's been given (ie: Paul knows Sam's been given the sum of the two numbers and visa versa)
Paul: I don't know what the two numbers are.
Sam: I know you don't know, neither do I.
Paul: AHA! I know what the numbers are now!
Sam: Ah, so do I


I'm not sure where to start. I know that there's a piece of mathematics that might be useful in this case: it's a rule that states that any given even number greater than 2 can be made by adding two prime numbers together.

Oh and: if you happen to figure it out, please don't spoil it for the rest of us, not to begin with at least

98C2= 4753 number of possible number combinations, so trial and error probably won't work for this

 

[TheCenturion]

Peter: I don't know what the two numbers are.
Simon: I know you don't know, neither do I.

* Peter takes out a pen and writes his number on his hand, as does Simon *

Peter: AHA! I know what the numbers are now!
Simon: Ah, so do I

I see you've used an exceptional amount of maths knowledge to reach that answer

 

[TheCenturion]

What's the point of this exactly if you can't tell anyone what the numbers are?

Just to begin with so that a fair few people can have a stab at it

 

[TheCenturion]

[
Are you being genuinely serious, or was that just two random numbers?

 

[TheCenturion]

Didn't realise that it was such a famous problem, only heard of it today


To anyone who's posted the answers: please could you edit your post so that anyone else who wants to figure out the answers can do so and read the thread at the same time?

Thanks

 

[TheCenturion]

I'll say it again: if you've posted the answer: PLEASE could you edit it out of your post, so as not to spoil it for other members?

 

[TheCenturion]

Sorry mate.

Done!

Thanks

 

[TheCenturion]

The most straightforward proof to this problem is computational (i.e. write a program).

I see the OP has been edited to remove the "what are the two numbers?" part of the question. Does this mean it is indeed impossible to narrow it down further, and we were just meant to explain Peter and Simon's reasoning?

It's entirely possible to do. I asked my head of maths today, and he said that he knew this particular problem. He knew 5 people who'd solved it (including himself) and all had used a different method. One of these people in fact had to write a program and left it overnight to figure out the answer

 

[TheCenturion]

Your maths teacher and his five friends seem to know more than the academics!

Actually, maths teacher was one of the five, so it's maths teacher + only four others that he knows who have ever solved this. One of the people who has solved it has now gone to Trinity (Cambridge) to do maths. He was literally a maths god. He took his GCSE in year 10, and at the time of doing his GCSE, he already had 8 maths A level modules under his belt

 

[TheCenturion]

I'm sure there is something to exclude these two pairs according to the problem statement but what is the flaw?

There is a flaw. It comes from the line:

I know you don't know what the numbers are

Any even number can be made from the sum of two primes. When two primes multiply together to make a product, the product can only have four factors: itself, 1, and the two primes.

Therefore, if the sum of the numbers was even, then there was a chance that both numbers could be prime numbers. However, this is not the case as he says with 100% certainty that he knows that the other person cannot know the two numbers. Therefore, the sum must be odd

 

[TheCenturion]

I've thought about that but can't we assume the guy said he doesn't know because he isn't sure? As the product can be made of either primes or non-primes so "not sure" equaled "don't know"? Maybe I'm getting too philosophical about it.

But the whole point is that if it is even, then there is a possibility that it could be made of two primes. However, the fact that it was said "I know you don't know what the two numbers are" means that the possibility could not have existed, therefore must be an odd number