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Great to Meet You! brainteaser

By , 22 June 2023

Difficulty: Tricky

You’ve just joined a new Scouts group! You meet 4 other new people, and everyone is going around giving handshakes. If everyone shakes hands with everyone else exactly once, how many handshakes will there be in total? (Remember you can’t shake hands with yourself.)

Scroll down or click for a hint, or the answer!

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Brainteaser hint

To get a start on this question, you could draw a diagram. Draw yourself and the 4 other people, and then draw lines between people to represent the handshakes.

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Brainteaser answer

There will be 10 handshakes in total.

One handshake is a combination of two people. This can be represented as a line between two points. In our drawing of the Scouts group, there are 10 unique lines (5 outer lines making the pentagon and 5 inner lines making the pentagram star).

Pentagon with star inside and letters A to E around the outside.

There’s another way to solve this question without a picture, but it’s easy to get wrong.

There are 5 people, and each person shakes 4 hands. 5 x 4 = 20 sounds like a good answer, but it’s wrong! We have to make sure we don’t double count a handshake between two people.

Writing out the handshakes shows how these duplications happen. The combinations in parentheses are combinations we’ve already written:

Table itemising handshakes between scouts.

Total handshakes: 4+3+2+1 = 10

There’s also a quicker way to remove the duplicates. Every handshake involves 2 people, and that means it’s counted twice. To get the correct answer, we divide by 2.

5 x 4 ÷2 = 10

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