[tombert]

When I was pretty young, I saw someone (I don’t remember who actually) show me that an odd number squared is always odd [1].

I remember thinking it was the coolest thing ever to be able to know properties for all numbers, without having to test any of them.

[1] most people here already know this, but for posterity, let’s define odd numbers as the expression 2k + 1. Now we square the expression which expands to 4k^2 + 4k + 1. We can factor this to 2(2k^2 + 2k) + 1. We can set the parenthetical expression to u, so we now have 2u + 1, which looks a lot like our original expression, showing it will always be odd.