We played the great game of SET and then answered these questions. The answers can be found in this document from The Madison Math Circle. Thanks to them and David J Bruce for the ideas.
We also started to think about how to represent cards in a geometric space. If you take a simple game of set with just two characteristics, you can plot each card in 2D and hence work out how many cards you can have on the table before you make a set (by not drawing a straight line). See here for the details.