Tri-puzzle: A three-cornered conundrum

Tri-puzzle is an interesting little puzzle in the spirit of packing problems popularized by Golomb but with an additional twist. It consists of sixteen triangular pieces that fit snugly into a triangular tray. Each of the pieces is marked with three colored spots, one on each side. The pieces must be packed in such a way that all adjacent spots within the tray are color matched. Also, spots on the outer edges of the tray must be color matched with the respective edge pieces within the tray. Lessons learned in the formulation of rectangular and cubic packing puzzles will help in modeling this puzzle but we run into a new and unexpected difficulty when attempting to generalize our solution to larger puzzles. A fiendish use of linear algebra allows us to surmount this obstacle.