Thank you to everybody for the great suggestions. I have decided what to do (see below) and I'll let you know how it goes after the picnic next week. I was inspired by the audience participation aspect of Josephus problem suggestion.
I will ask for a group of say 11 volunteers and ask them to shake hands with each other so that they all shake exactly three hands. Of course this is impossible and I will explain why. I may ask them if they can do this with an even number of people and have them try to come up with the construction.
Next I will describe a triple handshake (a group hug might be awkward...) and ask a group of 7 volunteers if they can do triple handshakes so that every pair of people are involved in exactly one triple handshake. Of course this is challenging without thinking about it for a while, but I will show them how to do it after they try. Then I will ask them if this can happen with 8 people. Again this is impossible and I will explain why. Now that we know there can be no "special triplehandshake system" (STS) on an even number of vertices I will explain why it can't happen with some odd numbers, like 11 for instance.