This problem is intriguing!

Number the sentences in conversation:
<<1>> B: I can’t tell what the numbers are.
<<2>> A: I can’t tell what the numbers are.
<<3>> B: I can’t tell what the numbers are.
<<4>> A: I can’t tell what the numbers are.
<<5>> B: I can’t tell what the numbers are.
<<6>> A: I can’t tell what the numbers are.
<<7>> B: Now I can tell what the numbers are.

Denote the sum S and sum of squares N. S-list will be a list of all possible sums a + b, N-list a list of all possible sums of squares a^2 + b^2. Initially, the S-list contains all positive integers >1, N-list all positive integers expressible as sum of two positive integer squares.

The section <<i>> below describes the information gained after the line <<i>> in the conversation was spoken.

<<1>> N is not expressible as the sum of positive integer squares uniquely. N-list is correspondingly adjusted (numbers expressible as sum of two positive integer squares uniquely are thrown out). S-list is adjusted – sums are thrown out, that don’t allow for summands a, b, such that a^2 + b^2 is on the current N-list.

<<2>> S allows for more values of a, b (a + b = S), such that a^2 + b^2 belongs to the N-list. S-list is adjusted (sums are deleted not allowing for more values of a, b (a + b = the sum), such that a^2 + b^2 belongs to the N-list).

<<3>> N allows for more values of a, b (a^2 + b^2 = N), such that a + b is on the S-list. Those integers that don’t are deleted from the N-list.

<<4>> S allows for more values of a, b, such that a^2 + b^2 belongs to the N-list. Those integers that don’t are deleted from the S-list.

<<5>> N allows for more values of a, b, such that a + b belongs to the S-list. Those integers that don’t are deleted from the N-list.

<<6>> S allows for more values of a, b, such that a^2 + b^2 belongs to the N-list. Those integers that don’t are deleted from the S-list.

<<7>> At this moment there is only one pair of positive integers a, b, such that a + b is on the S-list and a^2 + b^2 = N. If N-list contains more values, they need to be checked to see if they allow for unique solution.

If some bounds are imposed on a,b, I guess the solution could be found by computer or manual deleting from the S- and N-list. I haven’t really thought beyond this outline yet, at this moment solving the problem in full generality seems difficult… I’d love to see the intended solution.

SOURCE– http://www.artofproblemsolving.com/community/c146h150971