The sum of the first consecutive odd numbers is a square number.

Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.

Some of us (okay, me) may have presented something like this:

1 + 3 + 5 + … + (2*n* – 1)

(2*n* – 1) + … + 5 + 3 + 1

The sum of each column is 2*n*. We have *n* columns. The total is then *n *× 2*n *= 2*n*². We added the sum twice so 2*n*² ÷ 2 = *n*².

Can you see what perfect squares have to do with odd numbers? Me neither.

Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.

Inspired by this pictorial representation, I created this poster below.

¹ Lockhart might say it’s not the *fact* that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the* idea* of chopping the square into these nested shapes.