I have been trying to solve the following puzzle:

Take two integers n and m, selected from the range of integers 2 thru 100. Give the SUM of the two (n+m) to person S, and the PRODUCT of the two (n*m) to person P. Neither S or P know the values of m or n.

S says to P: "There is no way you can tell what my sum is"
P says to S: "Then, I know your sum"
S says to P: "Then, I know your product"

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

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!)

John Mann
jon.mann@btinternet.com