Nearly magic rectangles

MS Thesis Defense
Nearly magic rectangles

by Shara M. Baylin
MS Mathematics Candidate

Date: Monday, 23 May 2022
Time: 2 pm
Venue: Online

Advisers:
Dr. Jumela F. Sarmiento
Ateneo de Manila University

Panelists:
Dr. Jude C. Buot
Ateneo de Manila University
Dr. Eden Delight P. Miro
Ateneo de Manila University
Dr. Mark Anthony C. Tolentino
Ateneo de Manila University

Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to nearly magic rectangles of size p ×q, where p is even and q is odd. We prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist. Furthermore, we also consider the existence of a magic rectangle with empty cells of order m × n with precisely r filled cells in each row and s filled cells in each column, denoted by, MR(m;n; r ;s). We also construct nearly magic rectangles with empty cells of order m ×2m with 4 filled cells in each row and 2 filled cells in each column, denoted by, NMR(m;2m;4;2), for m = 3,5,7.

Key Words: nearly magic rectangles, magic rectangles with empty cells, nearly magic rectan- gles with empty cells