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Edit 3: 02/05/2020: It's starting to seem like it still might take the logicians quite a long time to figure out the numbers, and the logic gets increasingly difficult to follow. I wouldn't be surprised if for some pairs of numbers they might not be able to figure it out. Here's a particularly interesting (solvable!) case (shortened to aid readability). The numbers are 1 and 10:

P: I know you know that I don't know the numbers

S: I know you know that I didn't know that

P: I know the numbers now! They are 1 and 10!

Edit 2: As the condition about only talking about states of knowledge is ambiguous, I have decided that a new version is: The logicians can only say either they know/don't know the numbers, or that they knew/didn't know what the previous person just said.

Edit 2: As the condition about only talking about states of knowledge is ambiguous, I have decided that a new version is: The logicians can only say either they know/don't know the numbers, or that they knew/didn't know what the previous person just said (from which the other logician can infer that they don't know the numbers, because otherwise they would have said that)

Edit 2: As the condition about only talking about states of knowledge is ambiguous, I have decided that a new version is: The logicians can only say either they know/don't know the numbers, or that they knew/didn't know what the previous person just said.

For example, if the numbers are 3 and 4, a conversation could go:

P: I don't know the numbers (could be 1 and 12 or 2 and 6)

S: I knew you didn't (even if the numbers were 2 and 5 or 3 and 4, P wouldn't have known)

P: I didn't know that (S couldn't have said that if the numbers were 2 and 6. However, this doesn't yet eliminate 1 and 12)

S: I know the numbers! They are 3 and 4! (the previous statement eliminated the other options)