Darran A. Narayan
Rochester Institute of Technology

Mathematical Jigsaw Puzzles

Abstract:

Tilings problems are the jigsaw puzzles of a mathematician. For hundreds of years such problems have produced an array of beautiful and intriguing patterns. Problem B-3 from 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?" The answer is yes, and a solution appeared in the Monthly showing that L=2213 suffices. However, using an array of different algebraic and combinatorial tools, we will show that L can be significantly reduced to 33. 

*This is joint work with Rachell Ashley, Aisosa Ayela-Uwangue, Frances Cabrera, Carol Callesano, Grant Dietert,  and Allen Schwenk