Hard Tiling Problems with Simple Tiles

It is well known that the question of whether a given finite region can be tiled with a given set of tiles is NP -complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process we show that Monotone 1-in-3 Satisfiability is NP -complete for planar cubic graphs. In higher dimensions we show NP -completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply connected regions on the four-dimensional hypercubic lattice. Received March 8, 2000, and in revised form May 14, 2001, and June 18, 2001. Online publication October 12, 2001.