Alice knows a+b, Bob knows a×b. Can we find (a,b)?

Question

There are two distinct positive integers $a,b$ (i.e. $a\neq b$) such that $a < 7$ and $b < 7$.

Alice has been told the sum of $a$ and $b$. Bob has been told the product of $a$ and $b$.

Both Alice and Bob are aware of all the information mentioned above.

This is their conversation:

Alice says, "I am not sure if you know the sum that I have been told."

Bob says, "Actually, even before you spoke, I knew the sum that was told to you."

What are the possible values of $a$ and $b$ ?

Inspired by a Ted-ed video that I saw.

Answer 1

I believe the solution is

that the possible values of $(a,b)$ are $(1,4), (2,5), (3,5)$ (or swapping their orders).

First, we narrow down the solution space by observing that after Alice's statement

the only possible sums are $5, 7$ or $8$.

Then among these candidates, only the above solutions satisfy Bob's statement.

Detailed method

The following tables lists sums and the possible products corresponding to each sum. The starred products indicate that they appear in more than one row.

 sum=3: 2
 sum=4: 3
 sum=5: 4, 6*
 sum=6: 5, 8
 sum=7: 6*, 10, 12*
 sum=8: 12*, 15
 sum=9: 18, 20
 sum=10: 24
 sum=11: 30
 

We see that the only rows with a mixture of starred and nonstarred products are 5, 7, 8, which Alice observes. Within these rows, if Bob observes a non-starred product, then he may logically make his statement.