Adding 1 to 100 in seconds

Hello, and welcome to Myriora Maths! What is one, plus two, plus three, all the way up to one hundred? You could add for ten minutes. Or, you could do it in ten seconds.

The story goes that a young boy called Carl Friedrich Gauss was given exactly this problem at school. He had the answer almost at once. Here's how.

Write the numbers forwards. Then write them backwards underneath. Now add each pair. One and one hundred: one hundred and one. Two and ninety-nine: one hundred and one. Every pair makes the same total!

How many pairs are there? One hundred, one for each number. So both rows together add up to one hundred, times one hundred and one.

But that's two copies of our sum. So we halve it. One hundred, times one hundred and one, divided by two, is five thousand and fifty!

So here's the rule. Last number, times the next number, divided by two. One to ten? Ten times eleven is a hundred and ten. Halve it: fifty-five.

It works for any evenly spaced list, too. Two, plus four, plus six, all the way to forty. First plus last is forty-two. There are twenty numbers. Forty-two times twenty, divided by two, is four hundred and twenty.

Here's a common mistake: counting the numbers wrong. From two to forty, going up in twos, there are twenty numbers, not forty.

Your turn! Add up the numbers from one to fifty. You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! Fifty times fifty-one, divided by two. One thousand, two hundred and seventy-five!

Quick review. Pair the first number with the last. Every pair makes the same total. Multiply by how many numbers there are, then halve. See you next time!