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 m and n are integers greater than L, then an m × n 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:
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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 m and n are integers greater than L, then an m × n 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:
- Rachel Ashley
- Aisosa Ayela-Uwangue
- Frances Cabrera
- Carol Callesano
Return to NREUP Additional Information
Return to NREUP Home