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 4Ã?6 and 5Ã?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 4Ã?6 and 5Ã?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
4Ã?6 and
5Ã?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
4Ã?6 and 5Ã?7
rectangles. We will explore this problem as well as variations
involving
rectangles of varying dimensions.
Student Researchers:
- Rachel Ashley
- Aisosa Ayela-Uwangue
- Frances Cabrera
- Carol Callesano
Program Contacts:
Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473
Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470