>What are the values of n
>and m such that the
>above statements are true?

As you rightly note below the first statement, once uttered, becomes false, so that you have to rethink your formulation. But I understand what you mean.

>The appeal of this puzzle is,
>of course, that it appears
>to have insufficient information for
>the solution. Also, an intriguing
>point is that the first
>statement is true -- UNTIL
>S articulates it. Then it
>becomes false.
>
>The trouble is, I am not
>sure if it is possible
>to solve it. The first
>statement appears to be true
>for at least two answers,
>11 and 17:

It's true for 11, 17, 23, 27, 29, 35, 37, 41, 47, 53, 59, 65, 67, 71, 77, 79, 83, 89, 95, 97.

Call these numbers good sums.

>Suppose S has the sum as
>11
>the possible numbers are
>2+9 product=18, 3x6 or 2x9
>3+8 product=24, 12x2 or 3x8
>4+7 product=28, 14x2 or 4x7
>5+6 product=30 10x3 or 4x5
>
>
>so S can say "there is
>no way you can tell
>what my sum is"
>
>however for 17 this is also
>the case:
>
>2+15=17 6x5=30,10x3=30,15x2=30
>3+14=17 7x6=42,14x3=42,21x2=42
>4+13=17 13x4=52,26x2=52
>5+12=17 10x6=60,12x5=60,15x4=60,20x3=60,30x2=60
>6+11=17 11x6=66,22x3=66,33x2=66
>7+10=17 10x7=70,14x5=70,35x2=70
>8+9=17 9x8=72,12x6=72,18x4=72,24x3=72,36x2=72
>
>Am I missing something here, or
>is this really not possible
>to solve. (I *don't* have
>the answer!)

If you look at the products in your list and more generally for the products of two numbers that add up to a good sum, you may notice that some of them, like, e.g. 42, may be split into the product of two factors that add up to a good sum in more than 1 way:

42 = 14x3, 14+3=17, and
42 = 21x2, 21+2=23.

Or

72 = 24x3, 24+3=27, and
72 = 9x8, 9+8=17.

Call the products other than such good products. Far as I can see, there are only few good products:

18, 24, 28, 52 corresponding to the good sums
11, 11, 11, 17, respectively.

The first statement makes sense if S holds a good sum. The second statement is true only of P holds a good product.

The third statement may only be true if the good sum held by S is associated with a single good product. The only possibility that I can see is S = 17, P = 52, or m = 4, n = 13.

In all honesty, I have not verified the fact that there are only four good products. If it's not the case, the problem is unsolvable.