Here is a cool trick to square numbers made up of repeating 1 (one).

1. Count the number of repeating 1 in the number to be squared and put the digits ascending from 1 up to that number. That is, if there are four 1's in the number to be squared, put 1234.
2. Then, put down the same numbers inversely, that is, 321.

That’s it. It is so easy, it’s just mental math. Easy and fast.

Let’s illustrate with an Example:

If the number to be squared is 111

1. There are 3 repeating 1's. So, put down 123.
2. Then, put 21 after 123.

So, 111² = 12321

See the pattern?

Let's see another one, Square of 11111 =?

1. There are 5 repeating 1's. So, put 12345.
2. Then, put 4321

So, 11111² = 123454321

What happens past 9?

It works fine for 9,10,11,12,13,14 and 15 as long as you work in hexadecimal.

 

Incidentally if you want higher powers of 11 you can copy (at least the first of them) from Pascal's triangle.

http://en.wikipedia.org/wiki/Pascal's_triangle