I've devised a method for solving squares and square roots more rapidly by hand.

I noticed that the squares of integers always seem to increase by the sum of base of the integer you're solving for, and the base of the square previous to that.

For example 24^2 = 23^2 + 23 + 24

Integers can then be switched back to decimal form.

For example, 6.7^2 is first converted to the integer 67

We know 60^ = 3600
We then add 60+61 for the next square
And we add 61|62 for the square after that
And so on, until we reach 67^2

We then revert back to a decimal and voila. It may not seem impressive, but I have been able to solve for square roots this way in a matter of minutes, without any use of a calculator.

The Newtonian method of (a + b)^2 = a^2 + 2ab + b^2 is also a bit time consuming.