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Link Puzzle

Age 7 to 16 Challenge Level:

I was at a meeting with a colleague from the local Science Learning Centre and she had this metal "puzzle" on her desk.

I know I should have been focussed on what was being discussed at the meeting but I was distracted by the way the puzzle had been made and its mathematical properties. I hope this image helps you to see that the puzzle is made from sections of circular rings all joined so that each section can be rotated against the next.
Link puzzle

I first wondered how much each arc-shaped segment contributed to a full turn and so I tried to lay all the rings out flat...

Opening up the puzzle

and ended up with this:

Flattened puzzle

This certainly confirmed for me what angular contribution each section of the puzzle makes, and that got me thinking...

Firstly, about the number of sections that have to curve "in" and the number that have to curve "out". Look at the image and see if the answer in this case makes sense.

Secondly, I wondered whether there were other ways I could "flatten" the puzzle. It occurred to me that in fact this picture is a cheat. Can you explain why? You might find the interactivity below helpful or the arcs I created on paper here .

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Can you find any rules about the number of curves you need to make a closed arc? Are there any other properties of the resulting closed curves you can identify?

The puzzle was supplied by Frank Ellis of GlaxoSmithKline.