Two perfect logicians Steve and Pete, who have never met, before are imprisoned by an eccentric villain. "I have two positive integer numbers x and y" he says to them. "I will tell Steve the sum x+y, and Pete the product xy. If at any point either of you know the two numbers x and y, you may shout them out, and I will release you both. You may take turns saying statements giving the other information, but you may only communicate about your state of knowledge. If any part of your statement contains any information about the numbers themselves, you'll both be shot. If any of you tries to communicate any other information not directly implied by the statements, you'll both be shot."
Can the logicians always figure out the numbers?
As an example, if the two numbers are 1 and 6, a potential conversation could go:
P: I don't know the numbers
S: I don't know the numbers
P: I don't know the numbers
S: I don't know the numbers
P: I know the numbers! They are 1 and 6! (S would have known if the numbers were 2 and 3)
Or another example, if the numbers are 2 and 4:
P: I don't know the numbers
S: I didn't know whether you knew until you said that (could have been 1 and 5, and P would obviously have known, as 5 can only be factorized as 1x5)
P: I know the numbers! They are 2 and 4! (S couldn't have said that if the numbers were 1 and 8)
Is there always some strategy like this? (Note that the logicians are not able to communicate beforehand, and each logician must attempt to give useful information based on what they know)