That the other person is following the same system as them to decide whether they "know the numbers". It would be super easy for one of them to say they don't know for some other reason, in which the others' reasoning would be false.

This is what I thought. Two unstated (but implied) parts of the puzzle are that they can’t tell each other the numbers outright, and that they can only share information in yes/no format. But since they have clearly already agreed on a system beforehand, they could have also encoded the digits in binary or something.

So each person could have been sharing one binary digit of their number with each round. Or they might have used a completely different encoding. We simply cannot know.

That both are perfect logicians with an understanding of the system sans the information about the two numbers themselves?

since this is a fair assumption (that they will humanly miss a combination when true) the next inference is they have access to a perfect calculator

They both have to know what the other person knows (e.g the product or sum), and that the other person is a perfect logician.

Is it that the other person was told either the sum or the product? Or that the number they were told is the sum or product?

edit: After rereading the problem they weren’t told the parameters of the problem or even that they are participating in a puzzle.

process of elimination. as soon as peter starts, all primes are off the table. Given that, some number pairs which sum to the same number as just one other number pair where the second number pair has at least one prime are off the table. And so on and so forth. My instinct was to program the solution. Bet there are people who could do it in their heads.

It's only the primes above 50 that are immediately eliminated by the fact that Peter does not know the answer right away.

The primes below that can be present in products that have multiple possible answers: eg. we can't eliminate 47 right off the bat because Peter could have been told a product like 3290.

>as soon as peter starts, all primes are off the table

How do you figure? If Peter is given 97 (a prime), how is he supposed to know that a) he's been given a product (puzzle doesn't say he was told it's a product) and b) if he knows he's been given a product how is he supposed to know 97 is prime? He wasn't told that in the puzzle either.

The fact that it's unsolvable as stated is precisely why I hate this puzzle.

That the order of the numbers is unimportant? Otherwise every non-duplicate solution has a correlate solution, inverted. E.g. (3,2) implies (2,3) is also a solution.

I don’t think that’s necessary. The problem only says there are two numbers, not that the numbers have any order.

Each one of them:

1. is a person that solves logical puzzles if possible

2. knows that the other one + (insert point 1 here).

3. knows that the other one + (2)

4. knows that the other one + (3)

...

This is called Common Knowledge in game theory:

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum. It can be denoted as {\displaystyle C_{G}p}.

The number of steps it would take to solve the puzzle.

No, they discover that as they talk. It's not something they need to know; it's something you need to know.