Both answers are too complicated for me to understand. So I am going to use excel
Basically there are numbers that Bob cannot have. Bob cannot possibly have numbers like 100. Because if Bob have numbers like 100, then Bob would know the 2 numbers are and hence the sum. We need numbers for Bob that are products of multiple numbers between 1 and 10.
We need numbers that show up more than 1 on the above tables. Those numbers are
4,6,8,9,10,12,16,18,20,24,30,36,40 . Let's call them ambiguous product number.
All other numbers show up once.
So that's 0th step. Bob didn't know the number.
That means possible numbers that Bob may have is
Alice numbers can be
The first step is Alice knows that Bob didn't know. So Alice must have gotten a number where for all possible sum, Bob got product ambiguous numbers.
In other word, Alice got a number where all the diagonals are filled. Alice cannot possibly got 11 as a number. Because 11 can be 7+4. That means Bob may have 28 and 28 is not a product ambiguous number.
The only possibility is 5 and 7 (like what all other answers says)
So these are Alice possible numbers
Bob possible numbers are
Here number 6 shows up twice. So Bob's number can't be 6. We need numbers that show up only once. Basically Bob has a number where Alice's sum is obvious. So he needs a number that shows up once. That means Bob can have 4, 10, and 12
So Bob number can be 4, 10, and 12. Alice says now he knows the product. So by saying so, Alice reveals that her number is such that the down left to right top diagonal contains only one number. So we need to pick a diagonal with only one number. That means 4 (1, 4).
If Bob's number is 10 or 12, Alice number would be 7. But that means Alice wouldn't know the sum.
I think the first step, figuring out Alice sum can be simplified.
Bob has a number and Alice knows that Bob doesn't know her sum.
Hence, Alice sum would be x, where for all x, there does not exist a and b where a+b=x 0
So we need x, where for all a and b, if a+b = x, a*b is an ambiguous product number.
This is what's difficult to understand.
x cannot be 2. if x is 2, then a and b can be (must be in this case), 1 and 1 and that have a unique number.
x cannot be 3, because then a and b can be 1 and 2 (must be)
x cannot be 4, because a and b can be 1 and 3.
x must be something where all the sum have ambiguous factorization. Only 5 and 7 satisfy this.