I would like to square a three digit number in my head. Now I know that the formula is

$$ ( X + r ) ( X - r ) + r^2 = X^2 - rX + rX - r^2 + r^2 = X^2 $$ Where $\,r\,$ is a number such that $\,X + r\,$ is divisible by $10$ and/or $100$

Now the problem is that, I would like to first figure out whether finding the $r$ such that the $3$ digit number is divisible by $100$ or $10$ would be more efficient.

If I choose $r < 10$ then the $\,( X + r ) ( X - r )\,$ becomes quite unpleasant. If I choose $r < 100$ then the $r$ itself will have to be broken into pieces such that

$\begin{array}\\ ( X + r ) ( X - r ) + [ ( r + j ) ( r - j ) + j^2 ] &= ( X + r ) ( X - r ) + ( r^2 - jr + jr - j^2 + j^2 )\\ &=( X + r ) ( X - r ) + r^2\\ &= X^2 - rX + rX - r^2 + r^2\\ & = X^2\\ \end{array} $

This kind of yields recursive solution where any number squared works. The job is much more easier, but the person has to memorize more numbers while continuing to do more math.

So which method would you recomend? Or if you would have some other method I would love to hear it.

If you choose recursive method how do you memorize numbers in an efficient way without having numbers collide in your brain?

Thanks to anyone for their response, and sorry if tags are off, I am unsure which tag fits my question.