Latin squares and related combinatorial designs Leonard Soicher Queen Mary, University of London July 2013
Many of you are familiar with Sudoku puzzles. Here is Sudoku #043 (Medium) from Livewire Puzzles (http://www.puzzles.ca/sudoku.html): The aim is to complete the Sudoku, that is, to fill in the empty cells with numbers from 1 to 9 so that each number occurs once in each row and once in each column and once in each of the highlighted 3 by 3 subsquares.
And here is the completed Sudoku solving the above puzzle (again from Livewire Puzzles (http://www.puzzles.ca/sudoku.html)): Solving a Sudoku puzzle requires you to think mathematically.
The problem is stated precisely and has a precisely defined solution. You need to think logically and apply appropriate algorithms precisely. However, solving a Sudoku puzzle is different from research mathematics! A Sudoku puzzle is designed to have a unique solution. Often in research you don t know if there is going to be even one solution to a problem or many solutions. Sudoku puzzles are limited to small sizes (usually 9 by 9) and use limited shapes of subregions (usually subsquares), but mathematicians usually want to study a problem in greater generality, and obtain proved truths (theorems).
A completed Sudoku is an example of a combinatorial design, in which objects from a finite set are placed in a given structure so that given properties are satisfied. Combinatorial designs include mathematical objects called block designs, Latin squares, gerechte designs, mutually orthogonal Latin squares, SOMAs, orthogonal arrays, generalised t-designs, and many more. We will encounter some of these beasts in this lecture. Combinatorial designs are studied by statisticians for their use in the design of comparative experiments, and by pure mathematicians for their often beautiful structure and connections to finite geometry, graph theory (the study of connections and networks) and group theory (the abstract study of symmetry).
First, a completed Sudoku is an example of a Latin square of order n, which is an n by n array, with each cell of the array containing just one element from a given set of n symbols, such that each of the n symbols occurs once in each row and once in each column of the array. Next, a completed Sudoku is a special case of a gerechte design, which is a Latin square of order n whose cells are partitioned into n regions, each containing n cells, such that each symbol occurs once in each region. (Gerechte designs were introduced in 1956 by W.U. Behrens for the statistical design of agricultural experiments.)
Examples On the left is a Latin square of order 4. 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 5 6 7 8 7 8 5 6 8 7 6 5 6 5 8 7 On the right is a gerechte design. The regions are indicated by the colours, and are defined by the positions of the symbols in the Latin square on the left (so the white region consists of the cells where 1 occurs, the yellow region where 2 occurs, and so on).
When an n by n gerechte design can be made in this way, having regions defined by the positions of the symbols in a given Latin square of order n, we see that when we superimpose the Latin square of the gerechte design on the given Latin square, like so: 1 5 2 6 3 7 4 8 2 7 1 8 4 5 3 6 3 8 4 7 1 6 2 5 4 6 3 5 2 8 1 7 that the cells contain every possible pairing of a symbol from one square with a symbol from the other square (there are exactly n 2 such pairings), and we say that our two Latin squares are orthogonal.
A set of Latin squares of order n is called a set of mutually orthogonal Latin squares (or MOLS) when each pair of distinct squares in the set are orthogonal. For a given n, the maximum size that a set of MOLS of order n can attain is denoted by N(n). The example below illustrates a set of three MOLS of order 4, superimposed, and shows that N(4) is at least 3. 1 5 9 2 6 10 3 7 11 4 8 12 2 7 12 1 8 11 4 5 10 3 6 9 3 8 10 4 7 9 1 6 12 2 5 11 4 6 11 3 5 12 2 8 9 1 7 10
Some things known about N(n) For all n greater than 1, N(n) n 1. (This is not difficult to prove.) Whenever n is a power of a prime number, we have N(n )=n 1. (Thus N(2)=1, N(3)=2, N(2 2 )=3, N(5)=4, N(7)=6, N(2 3 )=7, N(3 2 )=8, N(11)=10, and so on. This result is proved by an algebraic construction using finite fields, which can only have prime-power size.) There is no pair of orthogonal Latin squares of order 6, so N(6)=1. (This was believed to be true by Euler, who discussed this problem around 1780, and a proof of this was finally published by G.Tarry in 1900.)
N(n) is at least 2 unless n=2 or n=6. (This was proved in 1960 by R.C. Bose, S.S. Shrikhande and E.T. Parker. Euler had conjectured that there were no pairs of orthogonal Latin squares of order n for n=2, 6, 10, 14, 18, and so on, but this turned out to be spectacularly wrong.) If n divided by 4 has remainder 1 or 2 and n is not the sum of two squares of integers, then N(n) < n 1. (This was proved by R.H. Bruck and H.J. Ryser in 1949. Thus, for example, N(14)<13.) Although 10=1 2 +3 2, N(10)<9, so the converse of the above result is false. (This was shown by C.W.H. Lam et al. in 1989, featuring much mathematics and very heavy computation. This was checked in 2011, by D. Roy, an MSc student at Carleton University, Ottawa.) Further results give better upper or lower bounds on N(n) for specific n. For example: N(10) 6, 5 N(12), 4 N(14) 10.
Some of the many things not known about N(n) The greatest unknown: Are there any n with n not a power of a prime, but with N(n)=n 1? The first open case is n=12. Indeed, for each specific n greater than 6 such that n is not a power of a prime, the exact value of N(n) is not known! Is there a set of three MOLS of order 10? (The problem is combinatorial explosion. Up to permutations of rows and columns and the naming of symbols, there are just 22 Latin squares of order 6, but the corresponding number for Latin squares of order 10 is 208,904,371,354,363,006.)
SOMAs: generalising sets of MOLS Let us return to the picture of three superimposed MOLS of order n=4: 1 5 9 2 6 10 3 7 11 4 8 12 2 7 12 1 8 11 4 5 10 3 6 9 3 8 10 4 7 9 1 6 12 2 5 11 4 6 11 3 5 12 2 8 9 1 7 10 Each cell contains the same number k=3 of symbols. Each of the kn=12 symbols occurs once in each row and once in each column, and each pair of distinct symbols occur together in a cell at most once (two symbols from the same Latin square never occur together and two symbols from different Latin squares occur together exactly once).
This motivates the definition of a SOMA(k,n), which generalises the concept of a set of k MOLS of order n. (SOMA stands for simple orthogonal multiarray.) Why do this? Mathematicians like to generalise. Designers of experiments may need (and have needed) designs that are like sets of k MOLS of order n, but where such a set does not exist or is not known to exist. SOMA(k,n)s have also arisen in other contexts, such as tournament design and message authentication. SOMA(k,n)s turn out to have nice properties.
Definition Let k>0, n>1. A SOMA(k,n) is an n by n array, whose cells each contain exactly k elements chosen from a given set of kn symbols, such that each of the kn symbols occurs once in each row and once in each column, and each pair of distinct symbols occur together in at most one cell. Note that a SOMA(1,n) is the same thing as a Latin square of order n and that a set of k superimposed MOLS of order n gives a SOMA(k,n).
A SOMA(k,n) is indecomposable if it is not the superimposition of two smaller n by n SOMAs. Example N(6)=1, but on the next page I show a SOMA(3,6), decomposed into a Latin square of order 6 and an indecomposable SOMA(2,6). (This particular design arose in a number of different contexts and was discovered or studied independently by a number of researchers, including R.A. Bailey, E.F. Brickell, N.C.K. Phillips and W.D. Wallis.)
Decomposable SOMA(3,6) with symmetry group of size 120 1 7 9 2 12 14 6 8 15 4 10 17 3 13 16 5 11 18 2 11 13 1 8 10 3 9 18 5 7 16 4 14 15 6 12 17 3 10 15 6 7 18 2 16 17 1 11 14 5 9 12 4 8 13 5 8 17 4 9 16 1 12 13 2 15 18 6 10 11 3 7 14 4 12 18 3 11 17 5 10 14 6 9 13 2 7 8 1 15 16 6 14 16 5 13 15 4 7 11 3 8 12 1 17 18 2 9 10
Some things known about SOMA(k,n)s Each SOMA(k,n) has a unique unrefinable decomposition into n by n arrays which are themselves SOMAs (proved by J. Arhin, who also gave an efficient algorithm to determine such a decomposition). If n is greater than 1, then k is at most n 1, and a SOMA(n 1,n) must be a superimposition of n 1 MOLS (proved by R.A. Bailey). A SOMA(n 2,n) must be a superimposition of n 2 MOLS (proved by J. Arhin); however, there exists an indecomposable SOMA(2,5), an indecomposable SOMA(3,6) and an indecomposable SOMA(4,7).
For n 6, the SOMA(k,n)s have been completely classified (done by me using my GRAPE and DESIGN packages in the GAP system for computational algebra and discrete mathematics). There exist both decomposable and indecomposable SOMA(3,10)s, an indecomposable SOMA(4,10) and indecomposable SOMA(4,14)s (constructed by me, using assumed symmetry, algorithms, and computation using GRAPE). Since it is known that N(n) is at least 3 when n>10, we now know that there exists a decomposable SOMA(3,n) for all n>3. On the next page, I show a SOMA(3,10), which is decomposed into a Latin square of order 10 and an indecomposable SOMA(2,10). On the page after that I show an indecomposable SOMA(4,10).
1 11 21 7 23 30 8 17 22 9 27 28 10 12 14 6 18 19 5 13 29 4 16 25 3 20 24 2 15 26 Decomposable SOMA(3,10) with symmetry group of size 10 2 3 4 5 6 7 8 9 16 27 12 26 19 28 17 23 13 14 18 20 24 25 15 30 1 2 3 4 5 8 9 10 12 22 17 28 13 27 20 29 18 24 14 15 11 19 25 26 9 1 2 3 4 5 10 6 21 24 13 23 18 29 14 28 11 30 19 25 15 16 12 20 10 6 1 2 3 4 5 7 18 23 22 25 14 24 19 30 15 29 12 21 20 26 16 17 6 7 8 1 2 3 4 5 28 29 19 24 23 26 15 25 20 21 16 30 13 22 11 27 7 8 9 10 1 2 3 4 13 15 29 30 20 25 24 27 16 26 11 22 17 21 14 23 8 9 10 6 7 1 2 3 19 20 14 16 21 30 11 26 25 28 17 27 12 23 18 22 5 10 6 7 8 9 1 2 14 30 11 20 15 17 21 22 12 27 26 29 18 28 13 24 4 5 7 8 9 10 6 1 17 26 15 21 11 12 16 18 22 23 13 28 27 30 19 29 3 4 5 9 10 6 7 8 11 25 18 27 16 22 12 13 17 19 23 24 14 29 21 28 10 22 29 6 16 21 7 26 27 8 11 13 9 17 18 5 12 28 4 15 24 3 19 23 2 14 25 1 20 30
1 2 19 20 17 30 31 32 10 11 36 38 9 12 15 29 8 33 35 40 5 25 34 39 4 6 16 21 7 18 23 37 13 22 26 28 3 14 24 27 Indecomposable SOMA(4,10) with symmetry group of size18 3 21 4 5 6 7 9 26 10 28 11 12 14 15 16 32 22 23 24 37 8 25 27 39 29 40 13 30 31 38 33 34 14 18 9 21 11 16 1 6 12 19 4 23 2 3 5 7 26 33 25 28 35 37 10 24 22 39 36 40 13 34 29 38 8 29 7 15 12 23 13 17 1 4 2 21 16 25 3 9 30 34 22 32 24 28 20 37 14 35 33 39 27 40 18 31 4 13 6 31 5 10 2 23 3 8 7 14 18 28 17 19 27 38 34 36 21 30 25 35 26 37 16 20 32 39 24 40 2 5 3 16 1 27 4 11 18 20 6 9 7 10 15 23 12 36 19 38 29 31 28 34 24 25 22 37 17 26 30 39 6 20 2 11 9 17 14 19 13 16 15 24 1 12 4 8 32 40 18 27 33 38 29 36 23 31 26 35 21 37 10 22 11 24 1 26 13 15 8 12 7 27 3 17 5 20 2 14 31 39 30 40 18 19 32 38 33 36 25 29 22 35 28 37 1 16 13 29 14 22 3 5 2 6 10 19 8 9 11 20 17 28 35 39 34 40 15 33 30 38 27 32 24 36 21 26 10 15 8 14 3 20 7 21 5 9 1 18 4 19 6 12 25 37 17 23 36 39 31 40 11 32 34 38 30 33 27 35 7 9 10 12 2 4 16 18 15 17 5 8 6 11 1 13 19 35 20 33 26 32 22 30 21 34 28 31 23 29 25 36 17 18 35 36 8 15 20 27 5 6 19 26 1 11 22 33 13 14 21 32 3 7 28 30 9 10 23 34 4 12 25 31 2 16 24 29 37 38 39 40
I would be very interested to see a SOMA(k,10) with k>4, a SOMA(4,18) or a SOMA(4,22). In addition, relatively little has been done to construct infinite families of SOMAs which are not superimpositions of MOLS. There is indeed scope for many new results. Thank you!