Maths

There are 6 people at a party. They all shake each other’s hands once. How many handshakes are there altogether?

At next week’s assembly, I’ll be demonstrating the classic “handshake problem.” When I asked for predictions from our volunteers, answers ranged from 30, to 25, to 18… and even 10!

When you try to solve it without a clear strategy, it quickly becomes confusing. It’s very easy to double count. For example, if Person A shakes Person B’s hand, and then you later count Person B shaking Person A’s hand, that’s the same handshake counted twice.

So how do we solve it properly?

One way is to think about it person by person:

• Person A shakes hands with 5 other people.
• Person B has already shaken Person A’s hand, so now shakes 4 new people.
• Person C shakes 3 new people.
• Person D shakes 2 new people.
• Person E shakes 1 new person.
• Person F has already shaken everyone’s hand.

So the total is:

5 + 4 + 3 + 2 + 1 = 15 handshakes

We can also write it more efficiently as:

6 × 5 ÷ 2 = 15

And for our algebraically inclined, this can be generalised as:

n(n − 1) ÷ 2

where n represents the number of people in the room.

Drawing a diagram can also be useful when solving problems like this. In the hexagon diagram provided, each person (A–F) sits at a vertex, and each line joining two vertices represents a handshake. Seeing all the connections visually helps ensure that each handshake is counted once, and helps us to look for patterns.

 We’ll unpack the thinking behind it at assembly on Monday 2 March.

Kelvin Tang

Mathematics Learning Specialist