Math Riddle, sum and product or integers larger than?

Question

Sam has a sum and Paul has a product of two integers both of which are larger than 1:

Sam says "Oy! Paul, You don't know my sum!"

Paul says "Yeah, you're right, and I still don't"

Sam say "That's alright I still don't know what your product is"

Paul says "I know your sum now!"

What are the two integers and how did you figure it out?

chancebeaube, if the sum were 8 sam couldn't say paul doesn't know his sum because it is possible that paul did (depending on what paul had)

Answer 1

Here is my reasoning so far, but I did not identify the numbers yet. See my concluding thoughts for where I would go next.


--------------------------------------------------
Sam has a sum and Paul has a product of two integers both of which are larger than 1:

Sam says "Oy! Paul, You don't know my sum!"
---------------------------------------------------
 Let A be the set of all such numbers that can be factored in more than one way by integers greater than 1.
Now let s = Sam’s sum. The sum could be from adding the integers 2 & (s-2), 3 & (s-3), 4 & (s-4), …
The possible values for p = Paul’s products are therefore 2(s-2), 3(s-3), 4(s-4), …
In order for Sam to be able to make his first claim, each of the values 2(s-2), 3(s-3), 4(s-4),… must be elements of set A. Let’s call this characteristic ‘the peculiar property’

---------------------------------------------------
Paul says "Yeah, you're right, and I still don't"
---------------------------------------------------
 In order for Paul to be able to make this claim, each possible factorization of p decomposes to numbers with pair wise sums that each have ‘the peculiar property’.

---------------------------------------------------
Sam say "That's alright I still don't know what your product is"
---------------------------------------------------
Okay, at this point I am not sure what to conclude from Sam's comment. The next step for me would be to construct set A and then the set of numbers with 'the peculiar property'. My suspicion is that only a few numbers actually have 'the peculiar property' and that by inspection, only one would leave Sam not knowing the final product.


Again, this is a great problem!

Answer 2

Let x>1, y>1 - integers
Let s = x+y - summ
Let p = x*y - product
Let PosP(s) - Set of all possibles p for given s
Let PosS(p) - Set of all possibles s for given p


Sam knows s. He write out all possibles for p (Set P).

Paul knows p. He write out all possibles for s (Set S).

>Sam says "Oy! Paul, You don't know my sum!"
PosS[p] for all p from P has more then one item.

Paul write out P1[s] for all s from S. It's possibles of sam's P.
And remove from S all items s, where exist n form P1[s], PosS[n] contain only one element.

>Paul says "Yeah, you're right, and I still don't"
Paul's S contain more then one element.

Sam write out S1[p] for all p from P (It's possibles of Paul's S), and each S1[i] process same way as Paul do.
After that he remove from P item p if S1[p] contain only one element.

>Sam say "That's alright I still don't know what your product is"
Sam's P contain more then one element.

Paul, for every s form S and every n from P1[s] write out S2[s][n] and process it like Sam's do at prev step.
Paul remove s from S if P1[s] contain only one element.

>Paul says "I know your sum now!"
Paul's S contain only one element.