Tiling problems have always been popular, especially after Hao Wang's non- recursive tilings of the plane, and Roger Penrose's non-periodic tilings. It is fairly easy to show that finite problems -- such as whether grid-based tiles can ever tile a square region -- are NP-complete. The author recently wondered whether a FIXED set of tiles could be shown to have trouble in tiling more complicated regions. A reduction of 3-SAT to such a problem will be shown.