[HWR_14]

Think about it like this. There are two numbers, 1-99 choices. So there are 4545 possible pairs (since 2,5 and 5,2 are the same we ignore order). However there are only 197 sums (2-198) and only so many products (I don't want to do the math on that, but obviously a number like 60 is reached by quite a few pairs). Each time one of them says "I don't know", the other considers every sum (or product) and asks if the other person has received enough information that that sum or product they know has one unique unelimiated pair that generates it. For some pairs (1,1) both players will have a unique answer right away. Otherwise, both players eliminate from the set of possible answers any pair that would compute an answer that no other non-eliminated pair would compute. That means the next time the player says "I don't know" they were doing so with a more constrained set of pairs. Which provides more information. Until eventually they eliminate all other possibilities to calculate their product/sum.

[at_a_remove]

I am still not getting this.

I think there is an assumption that both parties are ordering their possible choices in an identical manner, but I am unsure.

[HWR_14]

They aren't ordering it. Let's use a much simpler thing, the pair of numbers is 1-4. That gives us only 10 possible combinations. The product person says they don't know the answer. What do we know?

So if I told you the product of two numbers (positive integers) was 3, you immediately know 3 is prime and therefore the numbers are 1,3. Therefore either they cannot say they don't know if they were told 3. We can go through every combination and find what possibilities they could be.

Well, it cannot be 1,1 or 1,2 or 1,3 because those produce products of 1, 2 and 3 because they don't have any other pair that can generate them. The same is true of 2,3 which is the only way to get six or 2,4 the only way to get 8. And 3,3 or 3,4 which are the only way to get 9 and 12. And lastly 4,4 is the only way to get to 16. You'll note that this is almost all pairs which is easy to iterate over by making the second number greater than or equal to the first. The only pairs left are 1,4 and 2,2 both of which produce a product of 4. So we, as the sun person, now know the product must be four. Given the product and the sum, we can deduce that it's 1,4 because we were told the sun was 5.

When it goes to 100, it's the same process. Except now I have to iteratively eliminate pairs until there is only one solution.

[thfuran]

The only requirement is that they're both perfect (or at least sufficiently good) logicians and arithmeticians.

[at_a_remove]

In that case, I still don't get it.

[thfuran]

Alright, so at each step, they're trying to rule out as many wrong answers as they can until there's only one left, at which point they'll know the right answer. It works because they have different starting information and know the nature of each other's starting information, allowing them to combine information that they already know with the information communicated by the other person still not knowing the answer. Each time they say they don't know, that gives the other person some information—and they know they're giving each other this information. For example, say Sandy was told the sum is 84. When Peter says he doesn't know the answer, Sandy can immediately rule out (83,1) because that's the only pair of numbers with a product of 83. If Peter had been told that the product was 83, he'd have known the answer immediately. And Peter knows that Sandy can rule such pairs out. Every time Sandy says she doesn't know the answer, Peter can rule out from his remaining options the ones that he knows she has enough information to have known, had it been correct. And vice versa, repeating until one of them narrows it down to just one option.

[kadoban]

Is the basic idea clear? I'd say it's just that the statement "I don't have enough information" is _itself_ information that can be used to eliminate some possibilities.

After understanding that idea, the rest is just tedious logic/brute-force-search, I believe.

It's also possible that there is ambiguity in the statement or something like that. Hard to say exactly what part isn't connecting with you.

[at_a_remove]

Well, so they brute-force and come up with a set of possible answers, a list of tuples.

I don't have any guarantees that both of them are sorting each list the same. So how does "I don't have enough information" relay which of those list items to eliminate?

[ascar]

They always eliminate the pairs of products/sums with only one possible pair (think a sum to possible pairs and a product to possible pairs list).

When the person knowing the product says he doesn't know the answer, the sum person now checks the product list for all products with only 1 pair. These pairs can now be removed from the sum and product list. That actually culls the sum list and some entries that previously had 2 or more options now have 1 less. If the entry for the known sum has only 1 pair, he knows the answer. Otherwise he doesn't and now the culling continues with all sums that have only 1 pair. Repeat until you have the solution.

No ordering required. The algorithm can terminate at different rounds depending on the picked number and for some N (like 4) might not have a solution at all! For the given riddle the exact turn the solution was found is given through the conversation.

This is a pretty neat explanation https://alexanderell.is/posts/numbers-game/