Packing the maximum number of m × n tiles in a large p × q rectangle

Abstract For relatively prime m and n we determine precisely the wasted area when a large p × q rectangle is packed with m × n tiles in the most efficient manner. The case m = 1 is considered first, and we derive a formula for the wasted area depending only on the residue classes of p and q (mod n). This result has also been obtained by various other authors. Then by regarding an m × n tile as a union of 1 × n or 1 × m tiles a lower bound for the wasted area is obtained. Finally, by a series of explicit constructions, we show that for sufficiently large p, q, this lower bound is actually the correct value.