You have probably investigated how many handshakes it takes for everyone in a group to shake everyone else's hand once, and perhaps you have linked this with the 'mystic rose' investigation?

Have you ever investigated the number of regions or intersections inside the mystic rose? You could colour in the regions - how many colours do you need so that no two colours are touching? Perhaps you could now try planarity?

And... suppose people don't have to shake everyones hand... but can only shake peoples hands an odd number of times?  What happens? [hint: to make this interesting, choose an odd sized group!]