I think you are missing one more round. Impossible pairs after each round:

Round #1: ... (many)

Round #2: ... (many)

Round #3: (1,4), (72,92) and (72,98)

Round #4: (2,3), (80,90)

Round #5: (1,6), (75,96)

Round #6: (72,99)

Round #7: (81,88)

Round #8: (70,99)

Round #9: (77,90)

Round #10: (72,95)

Round #11: (76,90)

Round #12: (70,96)

Round #13: (90,84)

Round #14: (66,98)

Round #15: Solution is "77" and "84"