Two Integers greater than one.
Standard knows the sum.
Poor knows the product

Standard Says 'One Thing is certain, you cannot have determined my sum.'

Poor says after a suitable delay 'Thanks for telling me that but I
still don't know your sum'

Standard says after another delay 'And I don't know your product'

Poor says 'But now I know your sum'

What are the two integers?

Mongolia

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Request for Question Clarification by cynthia-ga on 08 Jun 2006 21:03 PDT

I don't want to spoil the fun simply because I can find the answer on
the internet --but I did.

Tell me when I can post it, for now I'll just bookmark it for later.

~~Cyn

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Clarification of Question by mongolia-ga on 09 Jun 2006 17:26 PDT

Cynthia

Before you formally answer my question, Please take note of MATHSISFUN last
comment. Was the question you saw on the Internet the same as the
question I posted?

Mongolia

Here's my guess: x=y=2. Reason: The cryptic information essentially
says that knowing the product or the sum one can obtain the other.
Probably this means that xy = x+y. Say y=x+n. Assume n>0. Then
x^2+nx=2x+n. But x^2>=2x and nx>n, contradiction. Therefore n=0 and
x^2=2x. So x=y=2.

I would disagree with Berkeley, if standard knows the sum is 4 and
poor knows the product is 4, then standard would not be able to say
'One Thing is certain, you cannot have determined my sum.' because
with integers greater than 1 the only thing that sums to 4 is 2 and 2
allowing standard to know that the product is 4 meaning that poor
would be looking at xy=4 x,y>1 which can of course only mean that
x=y=2 so poor would already be able to know standards sum making the
first quote false.

Product:30
Sum:17
Since poor is the solver of the game we look at it through his
perspective of having 30 with factor pairs of
2*15 meaning S has 17 as sum
3*10 meaning S has 13 as sum
5*6  meaning S has 11 as sum
When S says 'One Thing is certain, you cannot have determined my sum.'
This has to be taken as the fact that he has a sum which CANNOT be
made up of 2 primes, for it was then it would be possible poor would
know right of the bat the product of 2 primes cannot be divisible by
any other 2 numbers this means he cannot have 13 as a sum because that
could be expressed as 2+11.
Since P still has the possibility of the sum being either 17 or 11 he
needs to say 'Thanks for telling me that but I
still don't know your sum'

Now when S responds 'And I don't know your product'

We see that if he had 11 he would be able to verify our product because if we had:

2*9 = 18  with our possible choices for his sum being 11 or 9 we could
have ruled out 9 in his original statement so we would have known it
was 11 after his first hint

3*8 = 24 with our possible choices for his sum being 14 10 or 11 we
could have ruled out 14 and 10 in his original statement so we would
have known it was 11 after his first hint

4*7 = 28 with our possible choices for his sum being 16 or 11 we could
have ruled out 16 in his original statement so we would have known it
was 11 after his first hint

Leaving our only option to be 5*6=30 
However since he did say he didn?t know our product we need to verify
that there are 2 possible different products if S had 17 as the sum. 
One possibility would of course be 17=2+15 =>2*15=30 another would be
if P had 42 (3*14) then with all the given information P still
wouldn?t be able to narrow it down since S would then be able to have
either 17=3+14 or 23=2+21 (both of which I obviously chose since they
cannot be the sum of 2 primes)

Now since S said And I don?t know your product, which we?ve shown he
would had he had 11 as the sum but unable to with 17 as the sum and
those are our only 2 options we can positively state he has 11 as the
sum and we have 30

hmmm I see I was typing that for a while, well cyn, can you at least
tell me if I was close, not paying attention to my probably
incomprehendable use of run-ons?

I also found the answer on-line just now and it didn't match up, I'd
go through and try to find where I messed up but I have to do my own
homework (3 papers due tomorrow) silly college and it making math
majors take english courses, like I know how to read...

Nope, I can't tell ya, it's part of the game!  Plus, I found a page
that details all the places this question is found, all the variations
and answers are discussed. Upon a quick skim, I think there are 3
correct answers, and being right would mean you figured out there are
3 and what they are.

And I suspect you found the same or a similar page, because you are
right, your answer needs a bit of reworking. However, I think college
is more important than hanging out at GA waiting for interesting math
questions!

And you DO know how to read.

Hey Cynthia, did you find a page with the question as stated? I
noticed just know that the one I was looking at was
<<1>>   Mr. P.:  I do not know the two numbers.
<<2>>   Mr. S.:  I knew that you didn't know the two numbers.
<<3>>   Mr. P.:  Now I know the two numbers.
<<4>>   Mr. S.:  Now I know the two numbers.
which is a bit different the ours stated here.  when I worked the Mr.
P and Mr.S problem I got the correct answer.  I haven't gone through
all my math again as even I have trouble reading where I was going. 
Anyway, papers are done, no more classes for a month, and I get to
sleep for 2 horus before working for 8 (woo hoo) so I'll check back
tonight or tomorrow.

to MATHISFUN-GA

Just to let you know both your answer and reasoning by which you came
upon the answer agrees completely with my source.
i.e. Integers 2 and 15  (so sum is 17 and product is 30)

If you had been a researcher you would have been entitled to the sum of 17
dollars   
 (2 dollars for this question plus 15 dollars for the other question
making a total sum of 17 dollars :-)  of course minus the Google
Empire's commission)

So congratulations and even though I cannot give the princely sum of
17 dollars, I am sure this exercise was intellectually much more
simulating than your English Test.

Cynthia-ga

If you really did find this same problem on the WEB could you send me the link?

Regards to all

Mongolia

Thanks Mongolia, I had re-read my work and couldn't find any mistakes,
but goo do know for sure I was right, it was indeed a fun problem!