In an alternate universe, Peter and Sarah could have had the following conversation instead:

P: I don't know the numbers.

S: I knew you didn't know the numbers.

P: I knew that you knew that I didn't know the numbers.

S: I still don't know the numbers.

P: Now I know the numbers.

S: Now I also know the numbers.

But I'm worried that my version of the puzzle can no longer be solved without brute force.