After running this with my students, I had some really great ideas that the students generated on how to systematically keep track of the possible combinations of triangles, circles, and squares. The one that I liked the best at the end and that we all talked about together went like this.
Circle -> Triangle -> Square.
As the shape increased, so did the number of sides on the polygon (Circles had 1 side for this purpose… we had a long debate about that).
When we would increase past Square, the pattern would start over and the next most left column would increase by one in the shape pattern.
Essentially this was a base 3 counting system which I was able to re-emphasize with the Binary Odometer and making a base 3 as the last odometer (rather than base-20). In that case, 0 = Circle, 1 = Triangle, and 2 = Square.
I feel like the students took to it fairly well. I had a few groups try to figure out all of the base 4 problems by adding a star as an additional shape.