This project was quite an adventure from the beginning. While it was difficult, and ultimately impossible, to find the true formula that I was looking for at the beginning, I learned a lot about the implications of sequences and prime numbers in the grand scheme of mathematics. To recap briefly my project, I began with a numerical analysis of several different sizes of Latin square, including 3×3, 4×4, 5×5, 6×6, and 7×7. The number of possible Latin squares at each size increased rapidly, and after 7×7 it was much too large to realistically calculate. I was able to break these numbers down and apply a few different types of tests to these in order to find a pattern leading from one to the next, but the large primes which appeared in the prime factorizations of the numbers prevented any conclusive pattern from appearing. After spending a lot of time working with those numbers, I turned to the second stage of my project.
Computers can be extraordinarily frustrating. After completing the first stage of my project with some disappointment, I turned to the next stage, which was app development. The purpose of this was to create an app which would allow users to make their own Latin squares, and see how many distinct squares they could make on their own.
For the first part of my project, I attempted to find a pattern which could be followed in order to find the number of distinct Latin Squares that could be made at any given size. For more detail on the specifics of Latin Squares and the goal of this process, see my abstract here:
A Latin square is an n x n grid on which the numbers 1 through n are placed in each row, without any number being repeated in a row or column. An example of a 4 x 4 Latin square is given below. In this square, the numbers 1 through 4 are placed in each row and column with none being repeated.