dolem98

Ok if you insist, then maybe people should wait on the problem that I posted.

Lobstrosity

Well, I insist that my problem has a solution that meets my restrictions. I could have been mistaken in that the real problem places a cap on the sum and not the numbers themselves (I was trying to remember this from four years ago). At the very most, all capping the numbers at 50 would do is provide an extra solution, although I don't think this is a consequence. Similarly, all allowing the number 2 does is provide the possibility for extra solutions, and, in my mind, simply makes the problem harder by allowing you to exclude less right off the bat. All people should know is that there is a solution where Nick's numbers are both greater than two--find that solution.

Jesse

Lobstrosity, as far as I can tell I used exactly the steps you outlined in the gray text, but I didn't find any way for Sam to uniquely deduce the numbers after hearing Polly's answer. It might be that this would change if the upper limit was just on the sum or if 2 was allowed as a number, but as stated I think there is no answer to your problem. Do you remember the answer? If not, can you try to solve it yourself?

Silent Acorns

If you allow all numbers from 1 to 50 inclusive then there is a unique solution:

4 and 13

Jesse

Including the number 1? Wouldn't that mean almost every number from 1 to 100 is a possible sum, except for 2 and numbers of the form (1 + any prime under 53)?

Lobstrosity

I did. I solved it last night before I posted it to make sure I was remembering it correctly. I know the answer and I can demonstrate the solution. Should I just go ahead and do that?

Lobstrosity

Well, if you allow all numbers from 3 to 50 inclusive then there's a different solution. I have never worked through this problem allowing the number 2, so I cannot comment at this time on how that would change things.

Silent Acorns

This doesn't matter. Suppose we call the set of all combinations that allow Polly to figure out the answer after getting the info from Sam the "Polly Set". Since Sam knows the sum, he's looking for the only combination in the Polly set with this sum. For there to be a unique answer there has to be only one sum that has only one member in the Polly set. What lowering the minimum to 1 does is to make many of the product combinations in the Polly set non-unique (especially for the lower numbers, where any solution has to exist).

Silent Acorns

What's the solution then? I'm convinced that there isn't one to the problem you stated. Especially since sveral of us have come to the exact same conclusion, for the exact same reason, but using different methods.

Jesse

Silent Acorns:
This doesn't matter. Suppose we call the set of all combinations that allow Polly to figure out the answer after getting the info from Sam the "Polly Set". Since Sam knows the sum, he's looking for the only combination in the Polly set with this sum. For there to be a unique answer there has to be only one sum that has only one member in the Polly set. What lowering the minimum to 1 does is to make many of the product combinations in the Polly set non-unique (especially for the lower numbers, where any solution has to exist).

I'll take your word for it that you'll be left with only one sum that has a unique product associated with it, but still, with that many sums it seems like the only way to solve the problem would be to write a computer program to do it for you, since the number of possible sums consistent with Sam's first answer would be a little more than 80. It would be nicer if there was a version of the problem that you could figure out by hand (or with just a calculator) in a relatively short amount of time if you knew how to approach it.