Editing Packing problems (section)

==Related fields==
In tiling or [[tessellation]] problems, there are to be no gaps, nor overlaps. Many of the puzzles of this type involve packing rectangles or [[polyomino]]es into a larger rectangle or other square-like shape.

There are significant [[theorem]]s on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:
:An ''a'' × ''b'' rectangle can be packed with 1 × ''n'' strips if and only if ''n'' divides ''a'' or ''n'' divides ''b''.<ref name="Gems2">{{cite book | title = Mathematical Gems II | last1 = Honsberger | first1 = Ross | year = 1976 | publisher = [[The Mathematical Association of America]] | isbn = 0-88385-302-7 | page = 67 }}</ref><ref name="Klarner">{{cite journal | title = Uniformly coloured stained glass windows | journal = Proceedings of the London Mathematical Society |series = 3 | volume =  23 | issue = 4 | pages = 613–628 | last1 = Klarner | first1 = D.A. | last2 = Hautus | first2 = M.L.J | author-link1 = David A. Klarner | year = 1971 | doi = 10.1112/plms/s3-23.4.613 }}</ref>
:[[de Bruijn's theorem]]: A box can be packed with a [[harmonic brick]] ''a'' × ''a b'' × ''a b c'' if the box has dimensions ''a p'' × ''a b q'' × ''a b c r'' for some [[natural number]]s ''p'', ''q'', ''r'' (i.e., the box is a multiple of the brick.)<ref name="Gems2"/>

The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with [[congruence (geometry)|congruent]] tiles, and to pack one of each ''n''-omino into a rectangle.

A classic puzzle of the second kind is to arrange all twelve [[pentomino]]es into rectangles sized 3×20, 4×15, 5×12 or 6×10.