The basics are you break the group up into teams who design a "hotel" built out of snap cubes. These groups then earn income given one set of rules and are charged costs based on another (see rule sheets). I thought this would probably take the whole session but I printed an extra sheet of problems from the Julia Robinson Festival just in case. http://jrmf.org/problems/LittleBoxes.pdf It turns out the process took the entire hour as expected (and everyone had a really good time playing around with building designs). I reserved the last 10 minutes for scoring and if I repeat I'd bump that up to 15-20 minutes. The range in designs was really interesting and we didn't even probe all the optimization issues possible.
For the problem of the week I chose the current uwaterloo set which involves a number triangle from http://cemc.uwaterloo.ca/resources/potw/2015-16/POTWB-15-NN-24-P.pdf This is not too difficult and hopefully will lead to lots of students trying it out.
Looking forward my backup idea would make a great session on its own. I will definitely try out a mini Julia Robinson problem set in future weeks.