The rule of this puzzle is: The product of each of the above operations is equal to the last digit of the real product. 

Then:

1 × 1 = 01 (last digit is 1)

2 × 2 = 04 (last digit is 4)

4 × 4 = 16 (last digit is 6)

5 × 5 = 25 (last digit is 5)

7 × 7 = 49 (last digit is 9)

Thus:

9 × 9 = 81 (last digit is 1)

Hence, 9 × 9 = 1.