Complete latin squares of order 2k

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 145 (1995) 291-293

Note

Complete latin squares of order 2 k D . H. Griffel, a'* C. A. N . M o r r i s b aSchool of Mathematics, University Walk, Bristol BS8 ITW, UK b School of Industrial and Business Studies, University of Warwick, Coventry CV4 7AL, UK Received 26 November 1991; revised 4 January 1993

Abstract

The n x n matrix over Z, with (1,/) " " entry 5~ [l(l " " -- 1) + j ( j -- 1)] is a complete latin square if and only if n is a power of 2.

A latin square is an n x n a r r a y whose entries are integers in the set {0 . . . . . n - 1}, such that each n u m b e r in the set a p p e a r s once in each row a n d once in each column. A latin square is complete [1, 2, 5-1 if each pair of distinct n u m b e r s in {0 . . . . . n - 1} a p p e a r s as a pair of a d j a c e n t entries exactly once in some row a n d also in some column. There is a well-known c o n s t r u c t i o n which gives a c o m p l e t e latin square of any even o r d e r n (see [1-1). F o r some values of n, different squares can be f o u n d by a simpler construction; we shall p r o v e that this c o n s t r u c t i o n gives c o m p l e t e latin squares precisely when n is a p o w e r of 2. W e use the s t a n d a r d result that a c o m p l e t e latin square can be c o n s t r u c t e d from a sequencing of the set {0, 1. . . . . n - 1}. This m e a n s an o r d e r i n g of the numbers, say a~, a2,...a,, such t h a t the n p a r t i a l sums a l , a l + a2 . . . . . reduced mod(n), give all the n u m b e r s 0 . . . . . n - 1. T h e o b v i o u s o r d e r i n g is the n a t u r a l one with ai = i - 1 for each i. The following t h e o r e m gives the values of n for which this o r d e r i n g can be used to generate a c o m p l e t e latin square. Theorem. The natural ordering 0, 1,2 . . . . . n a p o w e r of 2.

1 is a sequencing if and only if n is

*Corresponding author. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 - 3 6 5 X ( 9 4 ) 0 0 0 4 1 - G

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D. H. Griffel, C. A. N. Morris~Discrete Mathematics 145 (1995) 291-293

Proof. The partial sums a r e ½r(r + 1), r = 0, 1,-.., n - 1. It is easy to show that they are all different mod(n) if n is a power of 2, so the natural ordering is a sequencing, and can be used to generate a square by the m e t h o d given below. N o w suppose that n is not a power of 2. Then n = 2kj for some o d d j / > 3. We show that there are two values of x in [0, n - 1] satisfying ½x(x + 1) = 0 mod(n), or equivalently,

x(x + 1) = 0 mod(2n).

(1)

Consider the two congruences

x(x + 1) = 0 m o d ( 2 k + l ) ,

x(x + 1) = 0 m o d ( j ) .

(2)

They are obviously satisfied by x - 0 and x = - 1. Hence if x=0mod(2

k+l)

and

x=

-lmod(j)

(3)

then (2) is satisfied, and therefore so is (1) because 2n = 2k+~j and the factors are relatively prime. The Chinese remainder theorem I-3, 4-1 guarantees a solution of (3); call it Xol. There is another solution X~o of (2) such that X~o = l mod(2 k÷~) and Xlo = 0 m o d ( j ) . It is easy to verify that 2 n - 1 - Xol = Xlo mod(2n). Thus Xo~ and X~o are symmetrically placed a r o u n d n - ½. Therefore one of them lies in [O,n - 1-1. But 0 is another solution of (1) in this interval. Thus the n partial sums are not distinct and cannot give all the numbers 0 . . . . . n - 1. Hence the natural ordering O, 1. . . . . n - 1 is not a sequencing. [] W h e n n = 2 k, a square can be constructed as follows. The first row is the sequence of partial sums ½x(x + 1)mod(n), x = 0 . . . . . n - 1. F o r i = 1 to n - 1 the (i + 1)th row is obtained by adding i to the ith row. The result is a complete latin square. It is easy to see that the entries of this square are as given in the abstract. W h e n n is not a power of 2, the p r o o f above shows that the first row of the square constructed by this m e t h o d contains two zero entries and it is not a latin square.

Acknowledgements We wish to thank Dr. H a r v e y Rose and an a n o n y m o u s referee for useful suggestions.

References [1] J. D6nes and A.D. Keedwell, Latin Squares and their Applications (English Universities Press, London, 1974).

D. H. Griffel, C. A. N. Morris~Discrete Mathematics 145 (1995) 291-293

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[2] J. D6nes and A.D. Keedwell, Latin Squares: New Developments in the Theory and Applications (North-Holland, Amsterdam, 1991). [3] G. H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (Clarendon Press, Oxford, 5th Ed. 1979). [4] H.E. Rose, A Course in Number Theory (Clarendon Press, Oxford, 1988). [5] A.P. Street and D.J. Street, Combinatorics of Experimental Design (Clarendon Press, Oxford, 1987).