This is part two of my weak attempts to solve ewd666 (go here to read part one).
This weekend I tried to attack the problem again and through some dumb luck I think I made some progress.
We left off in part one with the desire to find some property of S that will tell Sam that Paul cannot know the numbers. This time I went to hunt for facts of sums of two integers. The easiest way to hunt is of course with Google. After a few attempts with search terms that didn't yield anything interesting I tried it with "sum prime numbers" since prime numbers always play a big role in number theory problems.
Number one search result for "sum prime numbers" on Google is this page, a pretty interesting read. What caught my eye was Goldbach's Conjecture: every even integer greater than 2 can be written as the sum of two primes.
For our problem this means that S cannot be even. If S were even then Sam couldn't claim that he already knew that Paul cannot know the numbers because S = x + y with x and y prime so Paul could get the numbers from P = x * y.
It's a conjecture but hunting a little further reveals that it has been checked for all integers up to 6 x 10^16 so we're safe with our numbers in the range of [2..99].
This eliminates a lot of cases. I still feel there are way to many cases left so I'm not sure I made progress. It's hard to keep my curiosity in check and not go read the solution. I'm getting weaker, the temptation is great...
