Grant Dietert


We consider the problem of tiling large rectangles using smaller rectangles with the prescribed dimensions 4 x 6 and 5 x 7. Problem B-3 on the 1991 William Lowell Putnam Examination asked "Does there exist a natural number L such that if m and n are integers greater than L, then an m x n rectangle may be expressed as a union of 4 x 6 and 5 x 7 rectangles, any two intersect at most along their boundaries?" Narayan and Schwenk showed in 2002 that all rectangles with length and width at least 34 can be partitioned into 4 x 6 and 5 x 7 rectangles. We investigate necessary and sufficient conditions for an m x n rectangle to be tiled with 4 x 6 and 5 x 7 rectangles. Ashley et al. answered this question for all but 37 cases. We use an integer linear programming approach to eliminate all but five of these cases.

Library of Congress Subject Headings

Tiling (Mathematics); Linear programming; Integer programming; Computer algorithms; Rectangles--Mathematical models

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Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in December 2013. Physical copy available through RIT's The Wallace Library at: QA166.8 .D44 2010


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