School fair necklaces

How many possible symmetrical necklaces can you find? How do you know you've found them all?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Rob and Jennie were making necklaces to sell at the school fair.

They decided to make them very mathematical.

Each necklace was to have eight beads, four of one colour and four of another.

And each had to be symmetrical, like this.

 

Image
School fair necklaces



How many different necklaces could they make? 

Can you find them all?

How do you know you haven't missed any out?

What if they had 9 beads, five of one colour and four of another?

What if they had 10 beads, five of each?

What if...?