Many years ago, a family friend by the name of Pete showed me a couple math tricks. One was a way to remember the rules for right-triangle trig, and another was how to do a square root on paper. They used to teach you how to do it in school, you see. (Well, the square roots, at least.)
Many years after that, I got around to wondering about basic, mechanical processes for doing arithmetic. For example, multiplying out a couple numbers or doing long division. We're taught these things early in school, and never shown why they work -- they just work, we're to memorize the process, and then we take for granted that they work later in life.
This naturally led to wondering how the square root on paper algorithm worked. I sat down and figured it out, and made these pages as a result.
An algorithm for finding the square root of a number on paper is informally defined by way of this rather mechanical example.
For a better idea of what's actually going on, see the outline.
There's also a proof of the algorithm, which could use some cleaning up and clarification.
Finally, there's some observations about bases and an example of this algorithm in binary.