Overview
That 1 + 3 + 5 + … + (2n−1) equals n² is a fact you can literally see: each odd number is the next L-shaped layer of a growing square.
How to solve Odd Numbers Make Squares
- Start with a single cell — a 1×1 square, total 1.
- Wrap it with 3 cells (an L-shape) to make a 2×2 square: 1 + 3 = 4.
- Each new odd number 2n−1 is exactly the shell that grows an (n−1)² square into an n² square.
The key insight
The L-shaped shell added at step n has 2n−1 cells — one along the new column, one along the new row, minus the shared corner. So the odd numbers are precisely the square's growth increments.
Variations & echoes
- The same 'gnomon' idea gave the Greeks their figurate-number theory.
- Differences of consecutive squares (n² − (n−1)²) recover the odd numbers.
Frequently asked questions
Does it work for any n?
Yes. The first n odd numbers always sum to n², for every positive integer n.