Challenge Level

We can represent a group of friends by drawing a graph.

Each node represents a person.

An edge joins two nodes if and only if those two people are friends.

Here is a graph showing a group of friends.

**Can you work out who's who using the clues below?**

- Alan has 3 friends, Barney, Charlie, and Daniel.
- Barney and Ed are both friends with Charlie.
- Ed is Frank's only friend.

Here is a second network of friends.

**Again, can you use the clues below to figure out who's who?**

- Bella and Ciara are friends
- Emily and Ciara are not friends
- Bella is Fiona's only friend
- Anna has more friends than anyone else
- Daphne has three friends
- Gill and Daphne are not friends
- Emily has two friends

**Once you've solved the two puzzles, here are some questions to consider:**

Did each problem have a unique solution?

Were there any clues you didn't need to use?

If you label each node with the number of friends the person has, and add together all the numbers, what can you say about the answer? Can you explain why?

Can you design a puzzle which has a unique solution?

Can you design a puzzle which has two possible solutions?

Have a go at creating some other friendship network puzzles of your own and send them in for us to try!

*Download a printable version of this problem*

*This problem featured in an NRICH Secondary webinar in September 2021.*