half plane then it tiles the whole plane) and less so (for example, if a polyomino tiles an enlarged copy of itself, then it tiles the quadrant). Polyominoes of orders up to 6 are characterized in this hierarchy (with the status of one hexomino, later found to tile a rectangle, unresolved at that time). In 2001 Cristopher Moore and John Michael Robson showed that the problem of tiling one polyomino with copies of another is NP-complete. Tiling the plane with copies of a single polyomino Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes and all but four heptominoes tile the plane. It was then established by David Bird that all but 26 octominoes tile the plane. Rawsthorne found that all but 235 polyominoes of order 9 tile, and such results have been extended to higher orders by Rhoads (to order 14) and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them. The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to order 9 form a patch of at least one tile satisfying it, with higher-order exceptions more frequent. Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane. For instance, for every positive integer , it is possible to combine copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed. Tiling a common figure with various polyominoes The compatibility problem is to take two or more polyominoes and find a figure that can be tiled with each. Polyomino compatibility has been widely studied since the 1990s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results, and Livio Zucca shows results for some complicated cases like three different pentominoes. The general problem can be hard. The first compatibility figure for the L and X pentominoes was published in 2005 and had 80 tiles of each kind. Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible. Polyominoes in puzzles and games In