## A pictorial representation that will have you running naked through the streets

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 Â  Â  + Â  Â  3 Â  Â  + Â  Â  5 + â€¦ + (2n â€“ 1)
(2n â€“ 1) + â€¦ + 5 Â  Â  + Â  Â  3 Â  Â  + Â  Â  1

The sum of each column is 2n. We have n columns. The total is then nÂ Ã— 2nÂ = 2nÂ². We added the sum twice soÂ 2nÂ² Ã· 2 = nÂ².

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

Compare that with the following explanationÂ¹Â given in Paul Lockhart’s “A Mathematician’s Lament”.

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

Â Â¹Â 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.