We pick four numbers from 1 - 10.

Then we call 2 perfect logicians called Ann and Bob. They know we have four numbers but neither of them know the numbers.

We add those numbers and tell only Ann the result.

We multiply those numbers and tell only Bob the result.

Then we put them in a room together and ask them how many numbers do they know from the addition and multiplication of the numbers:

Ann: I don't know any number
Bob: I only know one number
Ann: I now know two numbers
Bob: I now know all the numbers!

What are the numbers?