What would be a good outdoor maths puzzle for children?

Solution 1:

If you want to have a topological outdoor game, involving 10-20 people, I know one, the gordian knot:

All participants come close together, rise their hands, and take firmly a hand of an other one in each left and right. This makes a terrific knot. The goal is to undo it, without letting go of hands.

Gymnastics, contortions, hughs, great laughs guaranteed ! In general, the knot becomes a simple loop, after about 15 minutes. Make statistics !

Solution 2:

I'm not really sure if this is a very good idea, but I'll just mention it anyway. The reason it might not be such a good idea is that I think the children could easily lose interest after less than 30 minutes, so you'd probably have to think about how to keep them focussed.

You could make a model of the Seven Bridges of Königsberg (e.g. with plastic sheeting for the river and planks for the bridges), tell the children the story of how people wondered whether it was possible to go for a walk around the city, crossing each bridge exactly once, and then ask them to either find such a walk or explain why it can't be done.
Perhaps each child could have some markers labelled 1 to 7 with their name or a unique colour or something, so that they can put down their marker whenever they cross a bridge. There should probably also be paper and pens available.

I think 20 is probably too many to do this at once though, unless your model is quite big.