Odd numbers make squares

Hello, and welcome to Myriora Maths! One, plus three, plus five, all the way up to nineteen. Here's a beautiful shortcut, and a picture that shows why it works.

Let's start small. One is one. One plus three is four. Add five: nine. Add seven: sixteen. Do you notice anything?

They're all square numbers! One squared, two squared, three squared, four squared.

Here's why. Start with one square. Add three, wrapped around the corner, and you get a two by two square. Add five more around the corner: three by three. Each odd number is the next L-shape that makes the square bigger.

So how many odd numbers did we add? From one to nineteen, there are ten odd numbers. And the answer is ten squared: one hundred!

A common mistake is squaring the last number. The answer isn't nineteen squared. Square how many odd numbers there are: ten.

Your turn! What is one, plus three, plus five, all the way up to twenty-nine? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! There are fifteen odd numbers, and fifteen squared is two hundred and twenty-five!

Quick review. Adding odd numbers from one always makes a square number. Count the odd numbers: the last one plus one, divided by two. Then square it. See you next time!