In the previous two parts (Part I and Part II), I showed you how to square three-digit numbers by splitting them up into hundreds and differences, and combining them in your head. So far so easy. But, you need to know your squares up to 50, and be happy multiplying - for instance $18 \times 49$ in your head. (It's 882, by the way).
Remember waaaay back, I talked about squaring numbers that end in 5? No? *sigh*. Well, let me recap. To square a number like 65, you multiply the first number (6) by the number after it on the number line (7) - $6 \times 7 = 42$ - and stick 25 on the end. $65^2 = 4225$.
One of the upshots of that is that 6.5 squared is 42.25, something you can use to your advantage when squaring numbers inconveniently far from a round hundred. Something like $648^2$, for instance.
I'd split that up as 6.5 hundreds and -2.
So, I'd work out 6.5 squared, which is 42.25, and move it left to get 4225.
I'd work out $6.5 \times -2 \times 2 = -26$ and add it on (4199).
And I'd square the -2 to get 04, giving me 419,904 altogether.
(One of the neat things is that you know you're going to double whatever half-humber you've got; if you do that straight away, it makes the sums easy. It's much easier to do $13 \times 2$ than $6.5 \times 4$!).
And that's it! All my tricks for squaring three-digit numbers. After that, it's all practice.