Can't tell quandry

Hint for Problem 4

("Know/Don't Know Game" Math. Mag. 1983, p. 177) After B's first announcement, the sum of squares must be attainable in at least two ways, and the possibilities are reduced to a table that begins as follows.

Sum Sum of Squares
8 50
9 65
10 50
11 65, 85
13 85, 125, 145
14 130, 170
15 125
16 130, 200
17 145, 185, 205
. . . . . .

Sum	Sum of Squares
8	50
9	65
10	50
11	65, 85
13	85, 125, 145
14	130, 170
15	125
16	130, 200
17	145, 185, 205
. . .	. . .

The only appearances of 50, 65, 85, 125, and 145 as a sum of squares in this table are already displayed. Keep going and you will find one possible answer. But how do you know it's the only possible one?

B's first announcement gives A a lot of information. But when A makes his first announcement, B already knows that A cannot possibly tell what the two numbers are. What good does it do B to listen to A?