Sad but true… I’m out of practice so I may have realized many times before but be too senile to remember (but maybe not, I could actually be dumb enough that this idea is new to me.).
I love my squares. 16*18 = 288 because 17^2= 289 - 1, done.
Yet when I can’t remember 23^2 I keep saying 20^2 + 2 * 20 * 3 + 3^2, or criss-cross. or skip a step 460+60+9… When I could just be saying 20 * 26 + 9 = 529. Funny how obvious that is. The other ways are fun but when resolving a square you are always within 4 of the 0, 5 doesn’t really count although with bigger numbers may you could lump it in 1295^2 , 1300*1200 + 25 seems pretty tidy (if a bit contrived).
I don’t think you would go wrong very often and find an easier method ( 1 digit x 2 digits + 1,4,9,16 ) for the first 99 squares. Ending in 0 and 5 are the only possible easier solutions assuming you have memorized a * (a+1) the n-1 number of digits that you are squaring kind of thing.