The handshake puzzle

Hello, and welcome to Myriora Maths! Ten people meet, and everyone shakes hands with everyone else, just once. How many handshakes are there?

Let's start small. Two people: one handshake. Three people: three handshakes. Four people? Let's draw them as dots, with a line for each handshake. Six!

Here's a clever way to count. Each of the ten people shakes hands with the nine others. Ten times nine is ninety. But every handshake has two people in it, so we've counted each one twice. Halve it: forty-five!

So the rule is: people, times one fewer, divided by two.

Look at the answers so far: one, three, six. Then ten, fifteen. These are the triangle numbers! And forty-five is one plus two plus three, all the way up to nine.

The puzzle often comes in disguise. In a league of eight teams, every team plays every other team once. Eight times seven, halved: twenty-eight matches. If they play home and away, don't halve: fifty-six.

The common mistake is forgetting to halve. Ten times nine counts every handshake twice.

Your turn! Twelve people all shake hands. How many handshakes? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! Twelve times eleven is a hundred and thirty-two. Halve it: sixty-six!

Quick review. Everyone shakes hands with everyone else. Multiply the people by one fewer. Then halve it, because each handshake was counted twice. See you next time!