Part 23. Latin Squares. Contents. 1 Latin Squares. Printed version of the lecture Discrete Mathematics on 4. December 2012

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1 Part 23 Latin Squares Printed version of the lecture Discrete Mathematics on 4. December 2012 Tommy R. Jensen, Department of Mathematics, KNU 23.1 Contents 1 Latin Squares 1 2 Orthogonal Latin Squares 3 3 MOLS 4 4 Conclusion Latin Squares Latin Squares Example: Every number 0,1,2,3 appears once in every row and column. Definition Let n be a positive integer. A Latin square of order n is an array A with n rows and n columns such that all entries are elements of {0,1,2,...,n 1} = Z n, and each element of Z n appears in every row of A, and each element of Z n appears in every column of A. 1

2 23.3 Using the pigeonhole principle, every element of Z n appears precisely once in every row and column of a Latin square A. Presentation of a Latin square If A is an n n array, then we write A = (a i j ) (0 i, j n 1). Then the rows of A are numbered 0,1,2,...,n 1 from top to bottom. And the columns of A are numbered 0,1,2,...,n 1 from left to right. The entry in row number i and column number j of a Latin square is the number a i j Z n. The array (a i j ) (0 i, j < n) is a Latin square if for every k Z n : for every i Z n there is a j Z n such that a i j = k, and for every j Z n there is an i Z n such that a i j = k Latin squares from modular addition Theorem Let Then A = (a i j ) is a Latin square of order n. a i j = i j (i, j Z n ). Proof of Theorem Let k Z n. Then for every i Z n we can choose j = i k, so that a i j = i ( i k) = k. And for every j Z n we can choose i = k ( j), so that a i j = (k ( j)) j = k. This proves that A = (a i j ) is a Latin square Latin squares from modular multiplication Theorem Let r be an element of Z n with a multiplicative inverse r 1. Define A = (a i j ) (i, j Z n ) by the rule: Then A is a Latin square of order n. Proof of Theorem Let k Z n. a i j = (r i) j (i, j Z n ). Then for every i Z n we can choose j = (r i) k, so that a i j = (r i) ( (r i) k) = ((r i) (r i)) k = k. And for every j Z n we can choose i = r 1 (k j), so that a i j = r (r 1 (k j)) j = (k j) j = k. This proves that A = (a i j ) is a Latin square. 2

3 This special Latin square has the name L r n Example of a latin square L r n Let n = 8 and r = 3. Then gcd(n,r) = gcd(8,3) = 1, this implies that r = 3 has a multiplicative inverse in Z 8. To calculate the entry a i j of L 3 8 for i, j Z 8 we have to calculate the remainder after division by 8 of So we get the rules, using addition in Z 8 : 3 i + j. the entry in the first row and the first column is a 00 = 0, we get the next entry to the right by adding 1, and we get the next lower entry of the column by adding L 3 8 Following these rules it is easy to construct L8 3 : Orthogonal Latin Squares Orthogonal Latin squares Definition Let A = (a i j ) and B = (b i j ) be Latin squares. They are called orthogonal Latin squares if they satisfy the following condition: For any two elements k and l of Z n, there exist i and j in Z n so that a i j = k and b i j = l. We can write another array (called the juxtaposed array) in which the position of row i and column j contains the pair (a i j,b i j ). Then A and B satisfy the condition for being ortogonal, precisely if each possible pair (k, l) of elements from Z n appear in this array

4 Example of orthogonal Latin squares and produce a juxtaposed array: (0,0) (1,1) (2,2) (3,3) (3,1) (2,0) (1,3) (0,2) (1,2) (0,3) (3,0) (2,1) (2,3) (3,2) (0,1) (1,0) We can check that all pairs of elements from Z 4 appear in the array MOLS Mutually orthogonal Latin squares Definition Let A 1,A 2,...,A k be Latin squares of order n. They are called mutually orthogonal if A r and A s are orthogonal for all r and s with 1 r < s k. A set of Mutually Orthogonal Latin Squares is called a MOLS. Theorem If n is a prime number, then L 1 n,l 2 n,...,l n 1 n form a set of MOLS with n 1 squares each of order n. Proof of Theorem We know from Theorem that L r n is always a Latin square of order n. It remains to prove that L r n and L s n are orthogonal for all r s with r,s {1,2,...,n 1} Proof that L r n and L s n are orthogonal for all r s. By definition of orthogonal Latin squares, we have to show, for each k and each l in Z n that (k,l) is in some entry of the juxtaposed array of L r n and L s n We know that L r n contains the number r i j in its i j-entry. And L s n contains the number s i j in its i j-entry. If we can find i and j so that r i j = k s i j = l are satisfied, then we know that (k,l) is in the i j-entry of the juxtaposed square

5 Proof that L r n and L s n are orthogonal for all r s, continued We want to find i and j to solve the equations r i j = k s i j = l Finding additive inverses of the second equation we get: We can add these two equations, and we get: r i j = k s i j = l r i s i = (r s) i = k l. Since r s, it follows that r s 0, and therefore the multiplicative inverse (r s) 1 exists in Z n, since n is a prime. Now i = (r s) 1 (r s) i = (r s) 1 (k l). From r i j = k we get j = k r i. We have now calculated the entry in which the pair (k,l) appears in the juxtaposed square from Ln r and Ln. s Constructing a MOLS of prime power order Theorem If p is a prime and n = p k for some positive number k, then there exists a MOLS of n 1 squares of order n. Proof of Theorem The proof is the same as for Theorem , using addition and multiplication of the finite field of order n = p k The maximal number of squares in a MOLS Theorem If a MOLS consists of squares of order n, then it has at most n 1 squares. Theorem (Tarry 1900) There are no two orthogonal Latin squares of order 6. Theorem For every odd number n, there exist orthogonal Latin squares of order n. Theorem (Parker, Bose and Shrikhande 1959) For every number n > 6 there exist orthogonal Latin squares of order n

6 4 Conclusion Conclusion This ends the lecture! Next time: Graph Theory

1 I A Q E B A I E Q 1 A ; E Q A I A (2) A : (3) A : (4)

1 I A Q E B A I E Q 1 A ; E Q A I A (2) A : (3) A : (4) Latin Squares Denition and examples Denition. (Latin Square) An n n Latin square, or a latin square of order n, is a square array with n symbols arranged so that each symbol appears just once in each row