Limits on the Ubiquity of Latin Rectangles

A year ago I investigated a mathematical problem relating to Latin squares. Most people, whether knowing it or not, have actually seen a Latin square at some point in their lives and many newspapers actually include partial Latin squares on a daily basis in the form of a sudoku puzzle. A Latin square is a grid of cells with numbers in each cell such that no number is repeated in any row or column, so any completed sudoku puzzle is really a 9x9 Latin square. Although Latin squares have been around for a while, providing entertainment in the form of puzzles to people ranging from Benjamin Franklin to high school students like me, there are actually quite a few open mathematical problems surrounding Latin squares. Latin squares have been used not only as puzzles, but also as tools to aid in eliminating bias in experimental design, and they are mathematically very interesting and have connections to areas like group theory and graph theory …

• Mathematics