This is why algebra is so useful. You noticed the pattern, but until you show it

*is* true in general you'd never be sure. (Your dad's proof is correct by the way.)

Actually, the reverse is often more helpful when dealing with algebraic matters (like trying to simplify equations). If you have the difference of any two squares, you can separate it because

.

BTW,

which can also be useful, but maybe not so often for mental arithmetic.

Edit: Also BTW, mental arithmetic tricks like this are often called number sense around where I'm from and

there are a lot more of them. They can all be broken down into algebra to prove they work, but some proofs can be surprisingly complex. (Sorry, no examples off the top of my head.)