There is a pattern when squaring "all 1-digited" numbers such as 1,11,111, and so on. It depends on the number of digits you are squaring.

For example, 11x11= 121

111x111=12321

1111x1111=1234321

As you can see, the digit-numbers increase consecutively from 1 to the number of digits you are using and then they decrease following the same pattern, so if you square an eight-digited "all 1" number, such as 11111111, you will get 123456787654321.