I recently came across a very easy way to find the squares of fairly large numbers ($1$-$100$, but could be extended indefinitely), by using a simple formula. It does require you to know the square numbers from $1$-$12$ but I'm sure that won't be a problem. Consider a number close to $50$ that you want to square, say $x$, then the formula goes like this:
For $x>50$ $$50^2+100(x-50)+(x-50)^2=x^2$$ For $x<50$ $$50^2-100(50-x)+(50-x)^2=x^2$$ i.e you add or subtract $100$ times the difference between $x$ and $50$ (depending on the size of $x$) and add the difference squared to $50^2$. Now this can be extended to close to $25,75$ and $100$ and beyond. For $25$ the only change is you add/subtract $50$ times the difference, for $75$ it's $150$ times, for $100$ its $200$ times and so on. Therefore you only need to know $(25k)^2$, $k\in\mathbb{Z}$ and the squares of $1$-$12$ since the number is always at most $12$ away from a multiple of $25$.
Some examples $x=36$ $$x^2=50^2-100(14)+14^2=2500-1400+196$$ $$=1296=36^2$$ $x=78$ $$x^2=75^2+150(3)+3^2=5625+450+9$$ $$=6084=78^2$$ It's a really neat trick with not much use as $x$ gets very large but something I thought I would share. It's especially useful if $x$ is extremely close to a multiple of 25. Has anyone else got any quick and easy mental mathematics tips to share? That could make life easier for a budding mathematician?
Thanks