The sum of the first n odd numbers is a square

Show that for every positive integer $n$,
$$1+3+5+\cdots +(2n-1)=n^2.$$

This is the friendliest thing on the board, and it is here deliberately. It has a proof by induction, a proof by pairing, and a proof that is a picture with no words in it at all.

Comparing those three is the quickest way to see what this board is actually trying to measure. They are all correct. They are not all equally illuminating, and they are not equally easy to check — and those two virtues pull in opposite directions more often than people admit.

If you are posting here for the first time, post here.