So I came across this neat pattern that I noticed, and so far I've only checked it for 5, and some two digit numbers that end with 5 and 3 digit numbers that end with 5.
The rule seems to be, if a number ends with 5, aside from 5 itself then the square will be a composition of 25 being the right most digits and then all digits after are equal to the square of the original number excluding the 5, plus that number again excluding the five. (below $ba$ is not a product)
So say $a$ will be the last two digits, and $b$ will be all of the other digits.
$$15^2=$$ $$a=5^2$$ $$b = 1^2 + 1$$ $$15^2 =ba= 225$$
$$2125^2$$ $$a=5^2$$ $$b=212^2 + 212$$ $$2125^2 = 4515625$$
Is this called anything in mathematics? Is there a way to prove this will be the behavior for all N digits numbers that end with 5.