Packing problems: Difference between revisions

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There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:

:An ''a'' &times; ''b'' rectangle can be packed with 1 &times; ''n'' strips iffif ''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'' &times; ''a b'' &times; ''a b c'' if the box has dimensions ''a p'' &times; ''a b q'' &times; ''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"/>