I like cats, I like group projects and I like puzzles (as a social activity).
I also like to cut stuff out using my scroll saw - though cutting lots of the same item sucks and would be MUCH easier with a laser cutter (pardon me while I wipe the drool off my chin thinking about an Epilog laser cutter :-).
This instructable describes how I made a set of cat puzzle pieces as a gift for my daughter's teacher. I made cats that interlocked, in a space filling, repeating pattern, called a tessellation. Good examples of tessellation can be found in the works of M.C. Escher - http://www.mcescher.com/
My daughter and her friends decorated lots of cats, which were then given to her teacher. We had a bunch of extra cats, which family members decorated and are pictured here. (Too bad I forgot to take a picture of ALL the cats that were given away - this was several years ago and the teacher has since retired).
Tessellations are wonderful for group projects because each person can decorate the tiles in a unique fashion, but they can also all be combined to make a larger focal piece. And, since all the pieces interlock, they can be assembled and reassembled in numerous variations.
Cat tessellations can also be hung together for a window/wall decoration, or, with added magnets, be fun fridge or white-board decorations. It might be fun to make a set of cat pieces for Halloween decorations.