what are A and B?tell me their value?

Problem Description

Two integers, A and B, each between 2 and 100 inclusive, have been chosen.

The product, A×B, is given to mathematician Dr. P. The sum, A+B, is given to mathematician Dr. S. They each know the range of numbers. Their conversation is as follows:

P: "I don't have the foggiest idea what your sum is, S."

S: "That's no news to me, P. I already knew that you didn't know. I don't know either."

P: "Aha, NOW I know what your sum must be, S!"

S: "And likewise P, I have figured out your product!!"

I saw this problem a long time ago, in the 1970's, and solved it. It's more tedious than hard. I'm not going to spend the time redoing it, but I recall what was involved in solving it.

You have to look at all the possible products and sums for number from 2 to 100, and figure out what they tell you E.g., if the product were 106, P would know that the numbers were 2 and 53, since these two primes are the only way to factor 106. Therefore he would know the sum, which is 55. So from his first statement you can rule out all the p x q products.

S's first statement seems to be misquoted. He says "I don't know either." But he DOES know the sum. I think the statement should be "I don't know the number given to you either."

When S says "I know that you didn't know" must mean that the sum can be formed in more than one way. So it's not something like 2+3, or 2+4, that can be formed from 2 different numbers in only one way.

When he says he doesn't know the product either, that gives P some information. Apparently from what P knows, there are several possible sums, all but one of which would have given S the product. Since S doesn't have the product, P can rule out some other numbers, and determine the sum.

S in term, learning this, now has enough info to narrow down the product.

It's complicated, and it was pretty timeconsuming. But I hope I've giving you a push in the direction of finding the answer.