Big project… might wanna start with the squares 11^2 - 99^2 which will already allow you to do the 2-digit criss-cross approach you’ve mentioned, albeit via difference of squares. On the other hand, that’s a lot of impact for just 90 numbers. You can then start adding the rest of them as needed.
I’ve just done exactly that (memorized the squares). I’ll probably write a post about the method I’ve used once I got a bit of free time…