# What would be a good outdoor maths puzzle for children?

I have to find an interesting activity for some 11-year-olds moving to high school this year. It is supposed to take about 30-45 minutes, and I thought of having a mathematical theme. I can make a large tangram, and do that. Pentominoes have some interesting puzzles attached. I could make a large Soma Cube - or get someone else to do it, but cubes of the right size to stick together seem hard to come by. There are some neat dissection puzzles, too. It would be possible to have several challenges done in rotation.

We're outside, and groups of up to 20, so there could be a competitive element. I want the thing big enough that if I split them into two or three subgroups they can all get involved.

The main thing is to have fun, but it would be great to get beyond that in a mathematical direction ...

All contributions welcome.

• It is not much of an activity, but I remembered this old video of solving the Steiner Tree Problem using soap bubbles youtube.com/watch?v=dAyDi1aa40E – utdiscant May 13 '12 at 19:54
• @utdiscant That's a great video - maybe not for what I wanted, but definitely for another day. – Mark Bennet May 13 '12 at 20:15

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.

• I think that's a fabulous idea, and could work really well on the site we have for our day. It would combine well with some other activities too. Thanks. – Mark Bennet May 14 '12 at 18:23
• My skepticism is because I have tried to do an activity on the Königsberg bridge problem before, and it didn't go so well. However, it was in a classroom with 14-year-olds rather than outdoors with 11-year-olds, so you might have better luck. Please let me know how it goes if you do go with this idea! – Tara B May 14 '12 at 18:47
• @Mark: I have some materials for this, which I could email to you if you like. (You can look up my email address here - I'm the only Tara there.) – Tara B May 14 '12 at 18:53
• I think this could work well as an outdoor activity: we're on grass, so some blue material and planks of wood should do the trick. It's not enough for 20, but I could do two others - one which is possible from anywhere, and one where you have to start and finish in the right place. And if I had a spare plank or two they could investigate how to make the task possible. And I could give them a sheet of paper with the graphs on at the end. The setup is a bit like "The Crystal Maze" with different kinds of activities - some physical, some mental etc. – Mark Bennet May 14 '12 at 19:31
• @Mark: Yes, doing three different ones as you suggested could work very well. The materials I was offering to send include a sheet of different graphs for the children to work out which ones can be drawn without taking the pen off the paper or backtracking, just to reinforce the ideas at the end. But you could make your own, of course. – Tara B May 14 '12 at 19:52

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 !

• +1 This is great (especially for group integration)! On the other hand, topology might be a bit too hard and abstract for high school. – dtldarek May 13 '12 at 20:28
• Also there is no guarantee that you'll end up with a single loop. – Dason May 13 '12 at 21:59

One idea which comes to my mind is to measure the height of objects as buildings or trees by measuring the shadow and using some gemetric theorems such as the intercept theorems. This obviously depends of the time of the day, and the theoretical background. Now that I think about it I don't really have a feeling anymore what 11 year olds know and this presumably also depends on the particular students and also the country you are from (i.e. the school system).

Another idea for something more competitive would be building paper planes and let them compete in different categories such as longest flight or farthest flight. Not sure whether this counts as mathematical, but the shape of the plane sure ought to be different for each different challenge. This needs some logical thinking.

I've seen Tantrix being played with large pieces outdoors. It's quite easy to construct the pieces, and alot af fun, especially if you race two different teams.

• That is a really good idea. – Mark Bennet Jan 11 '13 at 20:11

One example is a navigation. If you have a map and a compass (and maybe some friend does have a sextant), then there are multiple geometry problems you can devise, e.g.

• find where you are if you are given:

• two azimuths from known objects (towers, trees, originally lighthouses),
• two distances,
• angle between two objects (i.e. difference of azimuths) and distance from the first;
• if you know the height of an object, is it possible to know how far you are from it (e.g. for nautical navigation there are tables of lighthouses heights, etc.);

• how far you need to go to reach a place, where two points (e.g. trees) will be collinear, where is such point, is it convenient to go the shortest path (maybe there is a brook and you need to go over a bridge)?

It is even more fun, if you can draw a "secret map" with clues and hidden "treasure", for more mature kids orienteering might be a good idea (however this usually takes longer than 45 minutes).

After this there are many follow-ups, including celestial navigation (but it more fun on a boat/yacht), Earth coordinate system, GPS and Doppler effect, or even map projections.

Hope that helps!

You could have the children look at different natural objects and measure key ratios in their lengths and describe their self-similarities and so on. Turn it into a fractal expedition!

Just a comment now I've done this - I used the idea of graph tracing exercises on a large scale from Tara B.

I did three versions with different tasks.

The Königsberg bridge one I didn't make quite big enough, and my planks of wood need to look more like bridges.

Another was a puzzle to trace every edge on a map sourced from a collection of Dudeney puzzles. This too could have been bigger - the edges were 1 metre and could have done with being up to twice as long. This was made with blue rope, which was OK (but see below).

The final one, which worked best because it was biggest, was the graph of the edges of a dodecahedron, and find a hamiltonian cycle. The outer boundary was a pentagon of side approximately 3 metres - other edges approx 1 metre long - and the graph was made out of fluorescent yellow guy rope (which were usefully elastic) tied to curtain rings at the vertices. The vertices were held in place by tent pegs and this enabled the teams to trace out a cycle using a 30m rope with a loop at one end. Some markers were available to identify redundant edges as the cycle was traced out.

I found I had more space than I thought - so bigger and better next year.

I'm thinking about the next event now - and also about getting some resources together for a range of activities. JH Conway has some material on 'rational tangles' which could be quite good fun. I'd be grateful to hear about anyone who has tried this with children.

There are many to be honest. Like you said above, tangrams are definitely a great outdoor math puzzle for children and it's fun as well to engage with those 7 "tans". Then there are others like the Rubik's cube and all.But do you know that there are a few pen and paper games (Especially for children) as well that can serve this purpose effectively? Check these out-

Sink that ship

Aggression: An Amazing Pen/Pencil and Paper Game of Arithmetic