There are 4950 possible pairs in the initial problem statement. Sandy gets one of 197 possible sums, and Peter gets one of 2,869 possible products. Of those 2,869 products, 1,765 can be produced with only possible pair of numbers: something like 67 can only be (1, 67), whereas 240 could be (3, 80) or (5, 48) or (4, 60) or 5 other possible pairs. Peter doesn't know the answer, so when he tells that to Sandy, she learns that it can't be (1, 67) but it still could be (3, 80) or the like.
Before Peter told Sandy he didn't know, only 4 sums could have been caused by a unique pair (198, 3, 2, and 197). Peter telling Sandy he doesn't know lets her rule out lots of pairs, and after doing so, there are 9 sums that would have a single pair remaining that hadn't been ruled out. For example, were the sum 165, Sandy could have concluded that the only possible pairing would be 69 and 96, since the other pairs that add up to that number (e.g., 74 and 91, 80 and 85, etc.) would have unique products that Peter would have known about. That she doesn't know the answer yet therefore tells Peter that 69 and 96 is not a possible pair. Were the product 6624, Peter would now know that the only possible remaining pair was 72 and 92, and he would know the answer. But since he didn't know the answer, now Sandy knows that it can't have been 72 and 92 either.
This crossing out continues until Peter realizes that 70 and 96 was not a viable pair, which lets him realize that the only other way to get 6720 was to have the numbers be 80 and 84, and he declares he knew the answer. [Assuming I got the correct number of rounds]
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.
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.
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.
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.
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?
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.
Since Peter is given the product of the two numbers, he should instantly know the pair if both numbers are prime, but since he doesn't it rules out pairs like (7x11) = 77 and (2x53) = 106.
Sandy knows the sum and has now been told that Peter doesn't know the pair. If the sum had been 6, the following pairs are possible: (1+5) (3+3) (4+2), Peter has just ruled out (1x5) and (3x3), so Sandy would be able to narrow it down to (1,5) if the sum had been 6.
So when she tells Peter she can't narrow it down, it tells him that the pair isn't (4,2) either (among many others).
And if Peter's number were 8: (1x8) or (2x4) he'd be able to solve it, but he doesn't so Sandy then knows that (1,8) isn't the solution either.
>he should instantly know the pair if both numbers are prime
That should actually be "if the product of their respective smallest prime factors is over 100". 7x11 can't be ruled out since 1x77 also produces 77, whereas 49x17 and Nx53 can be ruled out.
You can also rule out some of the larger squares, e.g. 25x25 and 64x64, so there's probably still a better phrasing for that
Let's think about picking two numbers between 1-9.
Peter is given the product 24. He knows there are two possible pairs of numbers between 1-9 which produce a product of 24, (3,8) and (4,6), so he says "I don't know the numbers"
Sandy is given the sum 10. There are many pairs of numbers that produce a sum of 10, [(1,9), (2,8)...]. But she also knows that Peter did not immediately know the answer. If the pair of numbers was (5,5), that would have produced a product of 25. If Peter was given a product of 25, he would have immediately known the answer, since there's only 1 pair of numbers that produces that product.
So Sandy knows the answer isn't (5,5). Similarly, she knows it's not (2,8) or (3,7). The answer could be (1,9) though, since the product of (1,9) is 9, and there's another pair that can produce that product (3,3). If Peter was given the product 9 he wouldn't have immediately known the answer. The answer could also be (4,6), since the product of those is 24, and that can also be achieved with the pair (3,8). So there's only 2 pairs of numbers that add up to 10, and which Peter would not have immediately known based on their product. Sandy knows the answer must be either (1,9) or (4,6). Sandy says "I don't know the numbers".
Peter knows the solution must be either (3,8) or (4,6), and he knows that Sandy did not immediately know the answer. If Sandy had been given the sum 11 though, she should have immediately known the answer. There is only 1 pair of numbers that produces 11, but which does not have a unique product. Yes, the pair (2,9) sums to 11, but the product is unique, and if Peter had been given the product 18 to begin with, he would have immediately known the answer. So because he didn't immediately know the answer, and because that was not enough information for Sandy to say that the pair is (3,8), then Peter knows that the summation of the numbers must not be 11. The only other choice then is (4,6), and so Peter says "I do know the numbers".
> Yes, the pair (2,9) sums to 11, but the product is unique, and if Peter had been given the product 18 to begin with, he would have immediately known the answer.
pair (2,9) and pair (3,6) can produce a product of 18, so Peter can't immediately know the answer if he had been given the product 18 to begin with, so the problem can't beed solved.
I think this problem would Caught in an unresolved cycle after unique product and unique sum has been ruled out.
