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Intermediates Square Roots Sums of Squares Differences 
SquaresI'm sure you know this, but let me say it anyway. If you have a number n (1 2 3 4 … ), you can square it to get another number n^{2} (“n squared”, 1 4 9 16 … ). The concept is so familiar, you can almost forget where the name “square” came from.
I know a lot of squares reflexively, by association rather than calculation, but the upper limit varies. When I wrote Powers of N, the limit was 27; but right now it's 32, since I happen to know 28^{2} = 784, and the rest are easy. With a quick review I can push the limit up to 41, and with only a bit of effort I can push it to 50 or 55. That's a fairly recent development, though, helped along by learning the products up to 2100. Only the prime squares count as true products, but the other squares are useful as reference points; for example, 41×43 = 42^{2}1. Anyway, here's a table of all the squares I'm interested in.
Now, here's a funny thing. As long as I can remember, I've known the squares 24^{2} = 576 and 26^{2} = 676; I found them memorable because by strange coincidence they had the same last digits. Yes, believe it or not, it wasn't until a year or two ago that I noticed the pattern.
(25+d)^{2} = (25d)^{2} + 100d As d approaches 25, 25+d starts to look more like 50n.
(50n)^{2} = 2500  100n + n^{2} That's “another” pattern I'd never noticed, even though I've known about the striking number 49^{2} = 7^{4} = 2401 essentially forever. Finally, as you can see in the table, the pattern continues for numbers over 50.
(50+n)^{2} = 2500 + 100n + n^{2} The whole thing is really just a consequence of two little facts: (x)^{2} = x^{2}, and (x+2m)^{2} ≡ x^{2} mod 4m.

See AlsoAlgorithm, The History and Other Stuff Intermediates Logarithmic Forms Multiplication Table, The Powers of N Square Roots Sums of Squares Triangular Numbers @ July (2005) 