Trying again
Quote:
Let's call the product p, and the guy who has it P,
and the sum s and the guy who has it S.
For P to know the answer, p=axb
where a and b are primes ( not necessariliy distinct).
So if P doesn't know the answer, then a and b are not obviously primes.
For S to know that P doesn't know the answer, S must know that
no primes a and b exists such that s=a+b
The list of primes from from 2 to 98 inclusive are:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
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Here is where I made a mistake:
using the list we can calculate all the sums obtainable by adding two primes. There are only few sums not obtainable by that method. They are:
27 41 51 57 71 83 87 93 97 <--- wrong
11 17 23 27 29 35 41 47 51 57 59 65 71 77 79 87 89 93 95 97 <--right
So these are the possible sums.
But if a and b are the two numbers, then what is the minimum a x b for a + b that makes 97?
It's 95 and 2, but
95 x 2 = 190 which is bigger than 99
Only numbers less than 51 can be made by adding a to b, without a x b exceeding 99. So we can eliminate all numbers greater than 51. And we are left with:
11 17 23 27 29 35 41 47 51 <-- Our A list.
Those are the possible sums.
For 51 we get
2 x 49 = 98
If 98 is the product, it could be either 2 x 49 or 7 x 14
so P doesn't know the numbers. But for S to know that P doesn't know, it has to be 2 and 49. The sum of 7 and 14, i.e., 11, is not in our A list. So P and S both realise that the numbers are 49 and 2.
For 47
2 x 45 = 90
Hmmm... it could 2 and 45 too, 'cos 19 nor 21 are in our A list.
For 41
2 x 39 = 78
For 35
2 x 33 = 66
3 x 32 = 96
For 29
2 x 27 = 54
3 x 26 = 78
For 27
2 x 25 = 50
3 x 24 = 72
4 x 23 = 92
For 23
2 x 21 = 42
3 x 20 = 60
4 x 19 = 76
5 x 18 = 90
For 17
2 x 15 = 30
3 x 14 = 42
4 x 13 = 52
5 x 12 = 60
6 x 11 = 66
7 x 10 = 70
8 x 9 = 72
For 11
2 x 9 = 8
3 x 8 = 24
4 x 7 = 28
5 x 6 = 30
Stumped again. Is there more than one answer ?