Abstract:
In my thesis, I endeavor to provide a new, linear-algebraic proof of a classical theorem about
mutually orthogonal Latin squares. A Latin square of order n is an n × n array with n symbols
such that there are no repetition of entries among the rows or columns. Two examples of Latin
squares of order 4 are included in Figure 1. A Latin square is similar to a sudoku puzzle; however,
the sudoku has the added restriction of non-repetition in certain tiling regions.
0 1 3 2
2 3 1 0
3 2 0 1
1 0 2 3
0 3 2 1
2 1 0 3
3 0 1 2
1 2 3 0
00 13 32 21
22 31 10 03
33 20 01 12
11 02 23 30
Figure 1. Latin squares of order 4 and an array of ordered pairs.
To classify a pair of Latin squares as orthogonal, the ordered pairs created by superimposing
the squares atop one another must be distinct. Figure 1 gives a pair of mutually orthogonal Latin
squares of order 4 along with their array of distinct ordered pairs. A collection of Latin squares of
order n is said to be pairwise mutually orthogonal if each pair of Latin squares in the collection
is orthogonal. Such a collection is denoted MOLS(n).
These mathematical objects, Latin squares and MOLS(n), provide a means for conducting cer tain mathematical and experimental design projects. These squares can be utilized in block designs,
aiming experiments at comparing effects of different treatments or varieties. Utilizing these squares
in experimental design projects eliminates the biasing or confusing effect by randomizing and assign ing treatments to experimental units in a random manner. The concept of “blocking” in statistics
ensures independent variables are truly independent with no hidden confounding correlations. Thus,
the importance of my project which involves producing Latin squares reaches beyond the world of
mathematics and into the realm of data analysis - the field through which I am earning my degree.
These squares have real-life applications in experimental design problems I could potentially use in
my future career as a data analyst.
Much is already known about orthogonal Latin squares. Euler [3] was the first to consider the
orthogonality of Latin squares. This was done in the context of his 36 officers problem. In the 1700’s,
there was a common problem many attempted to solve involving all the Aces, Kings, Queens, and
Jacks of a standard card deck: One would attempt to create a 4 × 4 grid so each row and each
column contained all four suits as well as one of each face value. A similar problem faced Euler.
Though he was unable to solve the problem he conjectured that there were no orthogonal pairs of
Latin squares of order 4n + 2. This was proved in 1900 by Tarry [7] for n = 1, but was proved false
in 1959 by Bose, Parker, and Shrikhande [2] for n > 1.
The size of a collection of MOLS(n) is bounded above by n − 1. Bose, Moore, and Stevens
independently showed this bound is achieved when n is a prime power (see [1], [4], and [5]). A
classical proof of this result relies on manipulating the rows of finite field addition tables (via field
multiplication). It is this Bose-Moore-Stevens theorem that I wish to analyze in a new manner in
my honors thesis. Interestingly, if n is not 6 and not a prime power, then the maximum size of a
set of MOLS(n) is unknown.
This project is important for a variety of reasons, as it will give a tool that can be used to
study other, related objects, like sudoku-Latin squares. This project will also provide a new way
of looking at a classic theorem about Latin squares aimed at individuals who seek to further their
knowledge about these mathematical objects. This project is important to me as a student and
future businesswoman for many reasons. To begin, this process of independent research will help
me gain the time-management skills I will use in conducting future research projects throughout
my professional career. Further, this project will allow me to take a deeper look at the process of
creating unbiased research samples for experimental design projects. This is just the beginning of
my journey to research and research projects, given my future career path.