## Theses, Dissertations and Culminating Projects

12-2013

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Aihua Li

Jonathan Cutler

Mark Korlie

#### Abstract

A magic square of order n over a commutative ring R is an n x n matrix such that all the rows, columns, and the two diagonals add up to a fixed sum, which is called the magic sum. If all of the numbers in a magic square are perfect squares in R, it is called a magic square of squares. The rings under consideration in this thesis are either Z or Zp where p is a prime. In this thesis I present methods of constructing magic squares of squares of order 4 from selected ones of order 3. A natural question is, “Among those magic squares of order 4 which are constructed, what additional conditions are required so that all of the entries are perfect squares?”

In this thesis, Section 1 gives the history and background of magic squares. Simple examples of magic squares with all numbers distinct as well as examples of magic squares of squares with order 3 or 4 are examined. Basic number theory concepts are introduced in this section. In Section 2, I develop a method of constructing magic squares of order 4 from those of order 3 with magic sum equal to 0. I also identify which of those constructed are magic squares of squares. Section 3 shows the construction of magic squares of squares with 16 distinct perfect squares modulo a certain prime number p. In Section 4 we show that there exist infinitely many finite fields over which magic squares of squares with 16 distinct elements can be constructed. Section 5 focuses on developing methods for constructing magic squares of order 5 from magic squares of order 3. Conclusions are given in Section 6.

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