Graeco-Latin square

In combinatorics, a Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.

  • Graeco-Latin squares (Pairs of orthogonal Latin squares)
  • Order 3
  • Order 4
  • Order 5

The arrangement of the s-coordinates by themselves (which may be thought of as Latin characters) and of the t-coordinates (the Greek characters) each forms a Latin square. A Graeco-Latin square can therefore be decomposed into two "orthogonal" Latin squares. Orthogonality here means that every pair (s, t) from the Cartesian product S × T occurs exactly once.

History

Although recognized for his original mathematical treatment of the subject, orthogonal Latin squares have been known to predate Euler. As described by Donald Knuth in Volume 4A, p. 3 of TAOCP,[1] the construction of a 4 x 4 set was published by Jacques Ozanam in 1725 (in Recreation mathematiques et physiques, Vol. IV)[2] as a puzzle involving playing cards. The problem was to take all aces, kings, queens and jacks from a standard deck of cards, and arrange them in a 4 x 4 grid such that each row and each column contained all four suits as well as one of each face value. This problem has several solutions.

A common variant of this problem was to arrange the 16 cards so that, in addition to the row and column constraints, each diagonal contains all four face values and all four suits as well.

According to Martin Gardner, who featured this problem in his November 1959 Mathematical Games column,[3] the number of distinct solutions was incorrectly stated to be 72 by Rouse Ball. This mistake persisted for many years until the correct value of 144 was found by Kathleen Ollerenshaw. Each of the 144 solutions has eight reflections and rotations, giving 1152 solutions in total. The 144×8 solutions can be categorized into the following two equivalence classes:

Solution Normal form
Solution #1 A♠ K♥ Q♦ J♣
Q♣ J♦ A♥ K♠
J♥ Q♠ K♣ A♦
K♦ A♣ J♠ Q♥
Solution #2 A♠ K♥ Q♦ J♣
J♦ Q♣ K♠ A♥
K♣ A♦ J♥ Q♠
Q♥ J♠ A♣ K♦

For each of the two solutions, 24×24 = 576 solutions can be derived by permuting the four suits and the four face values independently. No permutation will convert the two solutions into each other.

The solution set can be seen to be complete through this proof outline:

  1. Without loss of generality, let us choose the card in the top left corner to be A♠.
  2. In the second row, the first two cells can be neither ace nor spades, due to being on the same column or diagonal respectively. Therefore, one of the remaining two cells must be an ace, and the other must be a spade, since the card A♠ itself cannot be repeated.
  3. If we choose the cell in the second row, third column to be an ace, and propagate the constraints, we get Solution #1 above, up to a permutation of the remaining suits and face values.
  4. Conversely, if we choose the (2,3) cell to be a spade, and propagate the constraints, we get Solution #2 above, up to a permutation of the remaining suits and face values.
  5. Since no other possibilities exist for (2,3), the solution set is complete.

Euler's conjecture and disproof

Orthogonal Latin squares were studied in detail by Leonhard Euler, who took the two sets to be S = {A, B, C, ...}, the first n upper-case letters from the Latin alphabet, and T = {α , β, γ, ...}, the first n lower-case letters from the Greek alphabet—hence the name Graeco-Latin square.

In the 1780s Euler demonstrated methods for constructing Graeco-Latin squares where n is odd or a multiple of 4.[4] Observing that no order-2 square exists and being unable to construct an order-6 square (see thirty-six officers problem), he conjectured that none exist for any oddly even number n ≡ 2 (mod 4). The non-existence of order-6 squares was confirmed in 1901 by Gaston Tarry through a proof by exhaustion. However, Euler's conjecture resisted solution until the late 1950s.

In 1959, R.C. Bose and S. S. Shrikhande constructed some counterexamples (dubbed the Euler spoilers) of order 22 using mathematical insights. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search on a UNIVAC 1206 Military Computer while working at the UNIVAC division of Remington Rand (this was one of the earliest combinatorics problems solved on a digital computer).

In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler's conjecture to be false for all n ≥ 10. Thus, Graeco-Latin squares exist for all orders n ≥ 3 except n = 6. In the November 1959 edition of Scientific American, Martin Gardner published this result.[3] The front cover is the 10 × 10 refutation of Euler's conjecture.

Mutually orthogonal Latin squares (MOLS)

A set of Latin squares of the same order such that every pair of squares are orthogonal (that is, form a Graeco-Latin square) is called a set of mutually orthogonal Latin squares (or pairwise orthogonal Latin squares) and usually abbreviated as MOLS or MOLS(n) when the order is made explicit.

For example, a set of MOLS(4) is given by:[5]

And a set of MOLS(5):[6]

While it is possible to represent MOLS in a "compound" matrix form similar to the Graeco-Latin squares, for instance,

1,1,1,1 2,2,2,2 3,3,3,3 4,4,4,4 5,5,5,5
2,3,5,4 3,4,1,5 4,5,2,1 5,1,3,2 1,2,4,3
3,5,4,2 4,1,5,3 5,2,1,4 1,3,2,5 2,4,3,1
4,2,3,5 5,3,4,1 1,4,5,2 2,5,1,3 3,1,2,4
5,4,2,3 1,5,3,4 2,1,4,5 3,2,5,1 4,3,1,2

for the MOLS(5) example above, it is more typical to compactly represent the MOLS as an orthogonal array (see below).[7]

In the examples of MOLS given so far, the same alphabet (symbol set) has been used for each square, but this is not necessary as the Graeco-Latin squares show. In fact, totally different symbol sets can be used for each square of the set of MOLS. For example,

Any two of text, foreground color, background color and typeface form a pair of orthogonal Latin squares:
fjords jawbox phlegm qiviut zincky
zincky fjords jawbox phlegm qiviut
qiviut zincky fjords jawbox phlegm
phlegm qiviut zincky fjords jawbox
jawbox phlegm qiviut zincky fjords

is a representation of the compounded MOLS(5) example above where the four MOLS have the following alphabets, respectively:

The number of mutually orthogonal latin squares

The mutual orthogonality property of a set of MOLS is unaffected by

  • Permuting the rows of all the squares simultaneously,
  • Permuting the columns of all the squares simultaneously, and
  • Permuting the entries in any square, independently.

Using these operations, any set of MOLS can be put into standard form, meaning that the first row of every square is identical and normally put in some natural order, and one square has its first column also in this order.[8] The MOLS(4) and MOLS(5) examples at the start of this section have been put in standard form.

By putting a set of MOLS(n) in standard form and examining the entries in the second row and first column of each square, it can be seen that no more than n − 1 squares can exist.[9] A set of n − 1 MOLS(n) is called a complete set of MOLS. Complete sets are known to exist when n is a prime number or power of a prime (see Finite field construction below). However, the number of MOLS that may exist for a given order n is not known for general n, and is an area of research in combinatorics.

Projective planes
Main article: Projective plane § Finite projective planes

A set of n − 1 MOLS(n) is equivalent to a finite affine plane of order n (see Nets below).[10] As every finite affine plane is uniquely extendable to a finite projective plane of the same order, this equivalence can also be expressed in terms of the existence of these projective planes.[11]

As mentioned above, complete sets of MOLS(n) exist if n is a prime or prime power, so projective planes of such orders exist. Finite projective planes with an order different from these, and thus complete sets of MOLS of such orders, are not known to exist.[10]

The only general result on the non-existence of finite projective planes is the Bruck–Ryser theorem, which says that if a projective plane of order n exists and n ≡ 1 (mod 4) or n ≡ 2 (mod 4), then n must be the sum of two (integer) squares.[12] This rules out projective planes of orders 6 and 14 for instance, but does not guarentee the existence of a plane when n satisfies the condition. In particular, n = 10 satisfies the conditions, but no projective plane of order 10 exists, as was shown by a very long computer search.[13]

No other existence results are known. As of 2020, the smallest order for which the existence of a complete set of MOLS is undetermined is thus 12.[10]

McNeish's theorem

The minimum number of MOLS(n) is known to be 2 for all n except for n = 2 or 6, where it is 1. However, more can be said, namely,[14]

MacNeish's Theorem: If is the factorization of the integer n into powers of distinct primes then

the minimum number of MOLS(n)

MacNeish's theorem does not give a very good lower bound, for instance if n ≡ 2 (mod 4), that is, there is a single 2 in the prime factorization, the theorem gives a lower bound of 1, which is beaten if n > 6. On the other hand, it does give the correct value when n is a power of a prime.

For general composite numbers, the number of MOLS is not known. The first few values starting with n = 2, 3, 4... are 1, 2, 3, 4, 1, 6, 7, 8, ... (sequence A001438 in the OEIS).

The smallest case for which the exact number of MOLS(n) is not known is n = 10. From the Graeco-Latin square construction, there must be at least two and from the non-existence of a projective plane of order 10, there are fewer than nine. However, no set of three MOLS(10) has ever been found even though many researchers have attempted to discover such a set.[15]

For large enough n, the number of MOLS is greater than , thus for every k, there are only a finite number of n such that the number of MOLS is k.[16] Moreover, the minimum is 6 for all n > 90.

Finite field construction

A complete set of MOLS(q) exists whenever q is a prime or prime power. This follows from a construction that is based on a finite field GF(q), which only exist if q is a prime or prime power.[17] The multiplicative group of GF(q) is a cyclic group, and so, has a generator, λ, meaning that all the non-zero elements of the field can be expressed as distinct powers of λ. Name the q elements of GF(q) as follows:

α0 = 0, α1 = 1, α2 = λ, α3 = λ2, ..., αq-1 = λq-2.

Now, λq-1 = 1 and the product rule in terms of the α's is αiαj = αt, where t = i + j -1 (mod q -1). The Latin squares are constructed as follows, the (i, j)th entry in Latin square Lr (with r ≠ 0) is Lr(i,j) = αi + αrαj, where all the operations occur in GF(q). In the case that the field is a prime field (q = p a prime), where the field elements are represented in the usual way, as the integers modulo p, the naming convention above can be dropped and the construction rule can be simplified to Lr(i,j) = i + rj, where r ≠ 0 and i, j and r are elements of GF(p) and all operations are in GF(p). The MOLS(4) and MOLS(5) examples above arose from this construction, although with a change of alphabet.

Not all complete sets of MOLS arise from this construction. The projective plane that is associated with the complete set of MOLS obtained from this field construction is a special type, a Desarguesian projective plane. There exist non-Desarguesian projective planes and their corresponding complete sets of MOLS can not be obtained from finite fields.[18]

Orthogonal array
Main article: Orthogonal array

An orthogonal array, OA(k,n), of strength two and index one is an n2 × k array A (k ≥ 2 and n ≥ 1, integers) with entries from a set of size n such that within any two columns of A (strength), every ordered pair of symbols appears in exactly one row of A (index).[19]

An OA(s + 2, n) is equivalent to s MOLS(n).[19] For example, the MOLS(4) example given above and repeated here,

can be used to form an OA(5,4):

r c L1 L2 L3
1 1 1 1 1
1 2 2 2 2
1 3 3 3 3
1 4 4 4 4
2 1 2 4 3
2 2 1 3 4
2 3 4 2 1
2 4 3 1 2
3 1 3 2 4
3 2 4 1 3
3 3 1 4 2
3 4 2 3 1
4 1 4 3 2
4 2 3 4 1
4 3 2 1 4
4 4 1 2 3

where the entries in the columns labeled r and c denote the row and column of a position in a square and the rest of the row for fixed r and c values is filled with the entry in that position in each of the Latin squares. This process is reversible; given an OA(s,n) with s ≥ 3, choose any two columns to play the r and c roles and then fill out the Latin squares with the entries in the remaining columns.

More general orthogonal arrays represent generalizations of the concept of MOLS, such as mutually orthogonal Latin cubes.

Nets

A (geometric) (k,n)-net is a set of n2 elements called points and a set of kn subsets called lines or blocks each of size n with the property that two distinct lines intersect in at most one point. Moreover, the lines can be partitioned into k parallel classes (no two of its lines meet) each containing n lines.[20]

An (n + 1, n)-net is an affine plane of order n.

A set of k MOLS(n) is equivalent to a (k + 2, n)-net.[10]

Transversal designs

A transversal design with k groups of size n and index λ, denoted T[k, λ; n], is a triple (X, G, B) where:[21]

  • X is a set of kn varieties;
  • G = {G1, G2, ..., Gk} is a family of k n-sets (called groups, but not in the algebraic sense) which form a partition of X;
  • B is a family of k-sets (called blocks) of varieties such that each k-set in B intersects each group Gi in precisely one variety, and any pair of varieties which belong to different groups occur together in precisely λ blocks in B.

The existence of a T[k,1;n] design is equivalent to the existence of k-2 MOLS(n).[22]

Graph theory

A set of k MOLS(n) is equivalent to an edge-partition of the complete (k + 2)-partite graph Kn,...,n into complete subgraphs of order k + 2.[10]

Applications

Mutually orthogonal Latin squares have a great variety of applications. They are used as a starting point for constructions in the statistical design of experiments, tournament scheduling, and error correcting and detecting codes. Euler's interest in Graeco-Latin squares arose from his desire to construct magic squares. The French writer Georges Perec structured his 1978 novel Life: A User's Manual around a 10×10 Graeco-Latin square.

See also

Notes
  1. The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Orthogonal latin squares.
  2. Recreation mathematiques et physiques, Vol. IV, p. 434, the solution is in Fig. 35
  3. Gardner 1966, pp. 162-172
  4. Euler: Recherches sur une nouvelle espece de quarres magiques, written in 1779, published in 1782
  5. Colburn & Dinitz 2007, p. 160
  6. Colburn & Dinitz 2007, p. 163
  7. McKay, Meynert & Myrvold 2007, p. 98
  8. Denes & Keedwell 1974, p. 159
  9. Denes & Keedwell 1974, p. 158
  10. Colbourn & Dinitz 2007, p. 162
  11. The term "order" used here for MOLSs, affine planes and projective planes is defined differently in each setting, but these definitions are coordinated so that the numerical value is the same.
  12. Bruck, R.H.; Ryser, H.J. (1949), "The nonexistence of certain finite projective planes", Canadian Journal of Mathematics, 1: 88–93, doi:10.4153/cjm-1949-009-2
  13. Lam, C. W. H. (1991), "The Search for a Finite Projective Plane of Order 10", American Mathematical Monthly, 98 (4): 305–318, doi:10.2307/2323798, JSTOR 2323798
  14. Denes & Keedwell 1974, p. 390
  15. McKay, Meynert & Myrvold 2007, p. 102
  16. Lenz, H.; Jungnickel, D.; Beth, Thomas (November 1999). "Design Theory by Thomas Beth". Cambridge Core. doi:10.1017/cbo9781139507660. Retrieved 2019-07-06.
  17. Denes & Keedwell 1974, p. 167
  18. Denes & Keedwell 1974, p. 169
  19. Stinson 2004, p. 140
  20. Colbourn & Dinitz 2007, p. 161
  21. Street & Street 1987, p. 133
  22. Street & Street 1987, p. 135

References
  • Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8
  • Dénes, J.; Keedwell, A. D. (1974), Latin squares and their applications, New York-London: Academic Press, p. 547, ISBN 0-12-209350-X, MR 0351850
  • Gardner, Martin (1966), Martin Gardner's New Mathematical Diversions from Scientific American, Fireside, ISBN 0-671-20913-2
  • McKay, Brendan D.; Meynert, Alison; Myrvold, Wendy (2007), "Small Latin Squares, Quasigroups and Loops" (PDF), Journal of Combinatorial Designs, 15 (2): 98–119, CiteSeerX 10.1.1.151.3043, doi:10.1002/jcd.20105}, Zbl 1112.05018
  • Raghavarao, Damaraju (1988), Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.), New York: Dover
  • Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.CS1 maint: multiple names: authors list (link)
  • Stinson, Douglas R. (2004), Combinatorial Designs / Constructions and Analysis, Springer, ISBN 978-0-387-95487-5
  • Street, Anne Penfold; Street, Deborah J. (1987), Combinatorics of Experimental Design, Oxford U. P. [Clarendon], ISBN 0-19-853256-3
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