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

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.