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Mutually Orthogonal Latin Squares and Their Generalizations Ghent University Faculty of Sciences Department of Mathematics Mutually orthogonal latin squares and their generalizations Jordy Vanpoucke Academic year 2011-2012 Advisor: Prof. Dr. L. Storme Master thesis submitted to the Faculty of Sciences to obtain the degree of Master in Sciences, Math- ematics Contents 1 Preface 4 2 Latin squares 7 2.1 Definitions . .7 2.2 Groups and permutations . .8 2.2.1 Definitions . .8 2.2.2 Construction of different reduced latin squares . .9 2.3 General theorems and properties . 10 2.3.1 On the number of latin squares and reduced latin squares . 10 2.3.2 On the number of main classes and isotopy classes . 11 2.3.3 Completion of latin squares and critical sets . 13 3 Sudoku latin squares 16 3.1 Definitions . 16 3.2 General theorems and properties . 17 3.2.1 On the number of sudoku latin squares and inequivalent sudoku latin squares . 17 3.3 Minimal sudoku latin squares . 18 3.3.1 Unavoidable sets . 20 3.3.2 First case: a = b =2 .......................... 25 3.3.3 Second case: a = 2 and b =3 ..................... 26 3.3.4 Third case: a = 2 and b =4...................... 28 3.3.5 Fourth case: a = 3 and b =3 ..................... 29 2 4 Latin squares and projective planes 30 4.1 Projective planes . 30 4.1.1 Coordinatization of projective planes . 31 4.1.2 Planar ternary rings . 32 4.2 Orthogonal latin squares and projective planes . 33 5 MOLS and MOSLS 34 5.1 Definitions . 34 5.2 Bounds . 35 5.3 Examples of small order . 46 5.3.1 Latin squares of order n ........................ 46 5.3.2 Sudoku latin squares of order n × n .................. 48 5.4 On the number of (n − 1)MOLS(n)...................... 49 5.4.1 First case: GF (p)............................ 49 5.4.2 Second case: GF (q)........................... 52 5.4.3 Example . 59 5.4.4 Further remarks . 62 6 Transversals 64 7 Magic squares 69 7.1 Definitions . 69 7.2 A little history . 70 7.3 Construction of magic squares . 72 7.3.1 First case: n =3 ............................ 72 7.3.2 Second case: n =4........................... 73 7.3.3 General case . 74 7.3.4 First case (bis): n =3 ......................... 74 7.3.5 Second case (bis): n =4........................ 75 7.3.6 General case . 75 3 Chapter 1 Preface Magic squares have long been an interest of mine and in my search to learn more about those magic squares, I stumbled upon latin squares. It fascinated me how it was possible that something which seemed so easy, could be connected with so many different and difficult areas of mathematics. Latin squares are not just some `spielerei' and I got the idea to write a thesis on this subject. On Tuesday, July 12, 2011, I had the opportunity to attend a lecture of Kenneth Hicks on the numbers of latin squares of prime power orders with orthogonal mates at the 10th International Conference on Finite Fields and their Applications in Ghent [15]. At the end of this talk, Kenneth Hicks conjectured that there are (p − 2)! distinct sets of (p − 1)MOLS(p), for p a prime, describing PG(2; p). This conjecture seemed to me an interesting challenge and my advisor, Prof. Dr. L. Storme, and I decided trying to prove this conjecture. This was not as easy as it seemed, but after a while we managed to prove the conjecture. An interesting idea that followed from this proof was to investigate the case in which we work with a prime power q = pd instead of a prime number p. Our goal was to find how many distinct sets of (q − 1)MOLS(q) there are, describing PG(2; q). This was not so easy and it took us some time to find the result. There was a lot of mathematics involved to achieve our goal and finally we found some new results, which are all presented in chapter 5. If we want to talk about mutually orthogonal latin squares, we will need some definitions and general theorems. These can be found in chapter 2. Another interesting subject are sudoku latin squares. We will see them as a special case of a latin square and more specific we will talk about a very interesting recent result of Gary McGuire on minimal sudoku latin squares in chapter 3, but first of all we will start this chapter with some definitions and some general theorems and properties of sudoku latin squares. Later on, when we will talk about mutually orthogonal latin squares, we will also discuss mutually orthogonal sudoku latin squares and we will prove the theorems found in [14]. An important chapter is chapter 4, because the connection between projective planes and latin squares is of great importance for our results on sets of MOLS. In chapter 6, we will briefly talk about transversals and finally we will discuss how we can construct magic squares by using mutually orthogonal latin squares. This thesis has not the intention to discuss everything that is known about the subjects. 4 This would be impossible, because there are a lot of theorems and properties known of latin squares, sudoku latin squares, MOLS; : : : However, these latin squares are very interesting and there still are a lot of open problems. The nice part is that new results on mutually orthogonal latin squares are connected to results on projective planes. Finally, I would like to thank some people for their support. First of all, I would like to thank my advisor, L. Storme, for his support and good assistance. It was not easy, but we achieved our goals and we found new results. Secondly, I would like to thank Kenneth Hicks and Gary L. Mullen for their ideas on sets of mutually orthogonal latin squares and Hans-Dietrich O. F. Gronau for his slides of his presentation in ALCOMA10, Thurnau, Germany, on orthogonal latin squares of sudoku type [14]. A special thank you to Tim Penttila for his great help concerning our questions on group theory. I would also like to thank my in-laws for their support. They always believed in me and they will always believe in me. Last but not least I would like to thank Domien Broeckx for his tremendous support. I appreciated all of the encouraging words. To end this preface I would like to say this. Whoever you are, reading this, I hope you will enjoy this thesis and I would like to thank you too for reading this. Jordy Vanpoucke Ghent, May 29, 2012 5 0 c by Jordy Vanpoucke The author and the advisor agree this thesis to be available for consultation and for personal reference use. Every other use falls within the constraints of the copyright, particularly concerning the obligation to specially mention the source when citing the results of this thesis. 6 Chapter 2 Latin squares 2.1 Definitions 2.1 Definition. A latin square of order n is an n × n matrix in which n distinct symbols from a symbol set S are arranged, such that each symbol occurs exactly once in each row and in each column. 2.2 Definition. We say that a latin square of order n is reduced (or in standard form), if the first row and the first column of this latin square is in the natural order of the symbol set we chose. From now on, we will work mostly with the symbol set S = f0; 1; 2; : : : ; n − 1g. An easy example of a reduced latin square can be obtained as follows. 2.3 Example. A reduced latin square of order n on the symbol set S = f0; 1; 2; : : : ; n−1g. 0 1 ::: n − 2 n − 1 1 2 ::: n − 1 0 . .. n − 1 0 ::: n − 3 n − 2 By giving this example we can see that the following theorem is true. 2.4 Theorem. There is a (reduced) latin square of order n for any positive integer n. Proof. We can take the integers 0; 1; : : : ; n−1 as the first row of an n×n matrix. To build the ith row, we simply do a cyclic shift of our first row, where we move the integers i − 1 positions to the left. By doing this we obtain for the ith row i−1; i; : : : ; n−1; 0; 1; : : : ; i−2. By construction, there are n distinct symbols and these symbols occur exactly once in each row. Also by construction, we can easily see that each symbol occurs exactly once in each column, so we constructed a latin square. Finally we see that the first row and first column is in the natural order of the symbol set, so we obtained a reduced latin square of order n. 7 2.5 Example. Another example of a latin square. 2.6 Definition. The transpose of a latin square L of order n on the symbol set S, denoted by LT , is the latin square defined by LT (i; j) = L(j; i) for all 0 ≤ i; j ≤ n − 1. 2.7 Definition. A latin square L of order n is symmetric if L(i; j) = L(j; i) for all 0 ≤ i; j ≤ n − 1. 2.8 Definition. A latin square L of order n is idempotent if L(i; i) = i for 0 ≤ i ≤ n−1. Now that we know from theorem 2.4 that we can find a latin square for any positive integer n, it would be nice to know how many distinct latin squares of order n there are for a positive integer n.
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