Hexaflexagons are a good experiment for young kids up to adults.

For the young kids you can treat it more as a logic puzzle; they need to figure out how to take a hexaflexagon and turn it inside out by only folding it (emphasise that you're only allowed to fold- you can do it by just 'pulling' the inside face through, but this is cheating and really damages the paper hexaflexagons). This (or other guides on the internet) show you how to do it correctly: http://www.puzzles.com/hexaflexagon/img/how_to_flex_a_trihexaflexagon.pdf

Once they've mastered that (get them to do it a few times), then ask them if they can do the opposite flex. See, there are 2 types of flexes, the one in the guide (the down flex) and the similar up flex. You do this by getting the other three corners and lifting them up to meet at the top, then opening out the bottom. Try to make sure they don't flip over the hexaflexagon as they do this. One way to encourage them to find the up flex themselves and not flip it over is to start the entire group off with their hexaflexagons with the same colours on the top and bottom as each other. Say green on top, pink on the bottom. Then ask them to, in one move, get pink on top and yellow on the bottom. Once they have up and down, if you still have there attention then you can get them to figure out equations: up up up= back to the start. flip up flip= down (this is a good one). If they're too young for that (or get confused with how many flips they've done- this happens a lot), move them onto the patterns.

A really fun one to get them to solve is, draw a sad face on a hexaflexagon (this has to be done in a particular way but there should be an example in the box). Ask them to figure out how to make it smile. This will illustrate to them that when you do one flex, the face that's still showing from before is now actually inverted. Give them each a blank hexaflexagon and get them to decorate it with this in mind- lines from the centre out and circles in the middle etc look really cool when flexed. There should also be one with pictures on it like a snake and a planet etc. Show them these for them to get some ideas for designing their own.

For more capable people, you can start talking to them about equations and even solving equations. For example, if any equation like (up flip up up) x (down down flip)=flip you can solve this. But to work up to this, get them used to the idea that ax=b means x=a^-1 b not x=ba^-1. Explain it as, order matters. Putting an action on the left means you do it last, where as putting it on the right means do it first.

----------Advanced--------------

Obviously, all Vihart's videos on Hexaflexagons are compulsory watching: https://youtu.be/VIVIegSt81k

The group here has 6 elements, and is in fact D_3 (or some people call it D_6). For very advance and capable students, show them that the hexaflexagon group is actually isomorphic to D_3 by using the triangle in the box. Ask them which elements get identified and how many rules they need to show to show that's true.

If you want to talk about more group theory, there are lots of shapes in the platonic solids experiment. Think about symmetries of these shapes. We can discover the whole D family of groups. You could also use a whiteboard to discover the symmetric groups. From here you can talk about subgroups, you should see D_3 and S_3 are the same and D_4 fits 'nicely' inside S_4. In fact Cayley's theorem says every group is naturally a subgroup of a symmetric group.