Unless I'm misunderstanding, I think problem 3 has many solutions. Spoilers: First note that, below 16, 11 is the only sum which explains S's first statement:

Unless I'm misunderstanding, I think problem 3 has many solutions. Spoilers: First note that, below 16, 11 is the only sum which explains S's first statement: all possible pairs of numbers adding to 11 have non-unique products, accounting for S knowing P would not know the numbers. Then note that for 18 (9x2), 24 (8x3), and 28 (7x4), all other factor pairs for that product add to a non-11 number below 16. Those non-11 sums are ruled out by S's first statement, so P will know the sum is 11 by her second statement. Therefore (9,2), (8,3), and (7,4) all look like valid solutions, and it seems likely there are more. What am I missing?