Why 1 + 3 + 5 + ... = n²

2026-05-01

Add up the first n odd numbers. The total is always a perfect square. Why?

An n by n square is just a stack of L-shaped layers. The first layer is a single cell. The second layer wraps around it - one cell to the right, one diagonally, one above - three cells. The third layer wraps around that - five cells. Every new layer is two cells wider than the last, because it has to cover one more row, one more column, and the corner where they meet.

Count the cells layer by layer and you get 1, 3, 5, 7, ... Count them all together and you get n², because that is what the square is. The sum of the first n odd numbers is a perfect square for the same reason a square has four sides: it is true by what the shape is, not by anything you have to prove.

Read it both ways

Drag the slider all the way left to see 1 = 1². Move it one step: now you have a 2×2 square, and the bottom-and-right L wrapping around the original cell contains 3 new cells, so 1 + 3 = 4 = 2². Each step adds the next odd number and produces the next perfect square. The reverse is true too: every perfect square is the previous square plus an odd-numbered L.

The Greeks called these gnomons

A gnomon was originally the L-shaped pointer of a sundial. The Pythagoreans borrowed the word for any L-shape that, added to a figure, keeps it the same shape but bigger. An L of odd cells added to a square gives the next square. An L of even cells added to a 1×k rectangle keeps it a rectangle. The whole arithmetic of figurate numbers - triangular, square, pentagonal - is built on this one move.

1 + 3 + 5 + ... + (2n - 1) = n². Not a formula to memorise. A square, with its layers showing.