Rochester Institute of Technology

Title: RIT Undergraduate Research

Director(s): Darren A. Narayan, Department of Mathematics and Statistics

Email: dansma@rit.edu

Dates of Program: June 6 - July 15, 2005

Summary: We consider the problem of tiling large rectangles with rectangles having dimensions 4×6 and 5×7. Some examples of these tilling are given below. Problem B-3 on the 1991 William Lowell Putnam Examination asked: "Does there exist a natural number L, such that if n are integers greater than L, then an rectangle may be expressed as a union of 4×6 and 5×7 rectangles, any two of which intersect at most along their boundaries?" A proof was given by Klosinski et al. showing that L = 2214 suffices (Amer. Math. Monthly 99 (1992), 715-724). Narayan and Schwenk showed that L could be reduced from 2214 to 33 and that this value is optimal. Thus rectangles with a length and a width of at least 34 can be tiled with 4×6 and 5×7 rectangles. However the case involving rectangles with a dimension less than or equal to 33 is still unsolved. Once completed this problem would yield a definitive list of all rectangles that can be tiled with 4×6 and 5×7 rectangles. We will explore this problem as well as variations involving rectangles of varying dimensions.

Student Researchers:

Program Contacts:

Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473

Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470