Incombinatorics, twoLatin squares of the same size (order) are said to beorthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set ofmutually orthogonal Latin squares. This concept oforthogonality in combinatorics is strongly related to the concept ofblocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.

An older term for a pair of orthogonal Latin squares isGraeco-Latin square, introduced by Euler.

AGraeco-Latin square orEuler square or pair oforthogonal Latin squares of ordern over twosetsS andT (which may be the same), each consisting ofn symbols, is ann ×n arrangement of cells, each cell containing anordered pair(s,t), wheres is inS andt is inT, such that every row and every column contains each element ofS and each element ofT 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 thes-coordinates by themselves (which may be thought of as Latin characters) and of thet-coordinates (the Greek characters) each forms aLatin square. A Graeco-Latin square can therefore be decomposed into two orthogonal Latin squares. Orthogonality here means that every pair(s, t) from theCartesian productS ×T occurs exactly once.

Orthogonal Latin squares were studied in detail byLeonhard Euler, who took the two sets to beS = {A, B, C, ...}, the firstn upper-case letters from theLatin alphabet, andT = {α , β, γ, ...},the firstn lower-case letters from theGreek alphabet—hence the name Graeco-Latin square.

When a Graeco-Latin square is viewed as a pair of orthogonal Latin squares, each of the Latin squares is said to have anorthogonal mate. In an arbitrary Latin square, a selection of positions, one in each row and one in each column whose entries are all distinct is called atransversal of that square.^[1] Consider one symbol in a Graeco-Latin square. The positions containing this symbol must all be in different rows and columns, and furthermore the other symbol in these positions must all be distinct. Hence, when viewed as a pair of Latin squares, the positions containing one symbol in the first square correspond to a transversal in the second square (and vice versa).

A given Latin square of order n possesses an orthogonal mate if and only if it has n disjoint transversals.^[2]

TheCayley table (without borders) of anygroup of odd order forms a Latin square which possesses an orthogonal mate.^[2]

Thus Graeco-Latin squares exist for all odd orders as there are groups that exist of these orders. Such Graeco-Latin squares are said to begroup based.

Euler was able to construct Graeco-Latin squares of orders that are multiples of four,^[2] and seemed to be aware of the following result.

No group based Graeco-Latin squares can exist if the order is an odd multiple of two (that is, equal to 4k + 2 for some positive integerk).^[3]

Although recognized for his original mathematical treatment of the subject, orthogonal Latin squares predate Euler. In the form of an old puzzle involvingplaying cards,^[4] the construction of a 4 × 4 set was published byJacques Ozanam in 1725.^[5] The problem was to take all aces, kings, queens and jacks from a standard deck of cards, and arrange them in a 4 × 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 toMartin Gardner, who featured this variant of the problem in his November 1959Mathematical Games column,^[6] the number of distinct solutions was incorrectly stated to be 72 byRouse Ball. This mistake persisted for many years until the correct value of 144 was found byKathleen 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 twoequivalence classes:

For each of the two solutions, 24² = 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, because suits and face values are different.

A problem similar to the card problem above was circulating inSt. Petersburg in the late 1700s and, according to folklore,Catherine the Great asked Euler to solve it, since he was residing at her court at the time.^[7] This problem is known as thethirty-six officers problem,^[8] and Euler introduced it as follows:^[9][10]

A very curious question, which has exercised for some time the ingenuity of many people, has involved me in the following studies, which seem to open a new field of analysis, in particular the study of combinations. The question revolves around arranging 36 officers to be drawn from 6 different regiments so that they are ranged in a square so that in each line (both horizontal and vertical) there are 6 officers of different ranks and different regiments.

Euler was unable to solve the problem, but in this work he demonstrated methods for constructing Graeco-Latin squares wheren is odd or a multiple of 4. Observing that no order two square exists and being unable to construct an order six square, he conjectured that none exist for anyoddly even numbern ≡ 2 (mod 4). The non-existence of order six squares was confirmed in 1901 byGaston Tarry through aproof by exhaustion.^[11][12] However, Euler's conjecture resisted solution until the late 1950s, but the problem has led to important work incombinatorics.^[13]

In 1959,R.C. Bose andS. S. Shrikhande constructed some counterexamples (dubbed theEuler spoilers) of order 22 using mathematical insights.^[14] ThenE. T. Parker found a counterexample of order 10 using a one-hour computer search on aUNIVAC 1206 Military Computer while working at theUNIVAC division ofRemington Rand (this was one of the earliest combinatorics problems solved on adigital computer).

In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler's conjecture to be false for alln ≥ 7.^[15] Thus, Graeco-Latin squares exist for all ordersn > 1 exceptn = 2, 6. In the November 1959 edition of Scientific American, Martin Gardner published this result.^[6] The front cover is the 10 × 10 refutation of Euler's conjecture.

Extensions of mutually orthogonal Latin squares to the quantum domain have been studied since 2017.^[17]In these designs, instead of the uniqueness of symbols, the elements of an array are quantum states that must be orthogonal to each other in rows and columns. In 2021, an Indian-Polish team of physicists (Rather, Burchardt, Bruzda, Rajchel-Mieldzioć, Lakshminarayan, andŻyczkowski) found an array of quantum states that provides an example of mutually orthogonal quantum Latin squares of size 6; or, equivalently, an arrangement of 36 officers that are entangled.^[16][18][19]This setup solves a generalization of the 36 Euler's officers problem, as well as provides a newquantum error detection code, allowing to encode a 6-level system into a three 6-level system that certifies occurrence of one error.

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 ofmutually orthogonal Latin squares (orpairwise orthogonal Latin squares) and usually abbreviated asMOLS orMOLS(n) when the order is made explicit.

For example, a set of MOLS(4) is given by:^[20]

And a set of MOLS(5):^[21]

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 (seebelow).^[22]

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,

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

• the background color:black,maroon,teal,navy, andsilver
• the foreground color:white,red,lime,blue, andyellow
• the text:fjords,jawbox,phlegm,qiviut, andzincky
• the typeface family:serif,sans-serif,monospaced,cursive, andslab-serif.

The above table therefore allows for testing five values in each of four different dimensions in only 25 observations instead of 625 (= 5⁴) observations required in afull factorial design. Since the five words cover all 26 letters of the alphabet between them, the table allows for examining each letter of the alphabet in five different typefaces and color combinations.

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 intostandard 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.^[23] 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 thann − 1 squares can exist.^[24] A set ofn − 1 MOLS(n) is called acomplete set of MOLS. Complete sets are known to exist whenn is aprime number orpower of a prime (seeFinite field construction below). However, the number of MOLS that may exist for a given ordern is not known for generaln, and is an area of research incombinatorics.

A set ofn − 1 MOLS(n) is equivalent to a finiteaffine plane of ordern (seeNets below).^[10] As every finite affine plane is uniquely extendable to afinite projective plane of the same order, this equivalence can also be expressed in terms of the existence of these projective planes.^[25]

As mentioned above, complete sets of MOLS(n) exist ifn 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 theBruck–Ryser theorem, which says that if a projective plane of ordern exists andn ≡ 1 (mod 4) orn ≡ 2 (mod 4), thenn must be the sum of two (integer) squares.^[26] This rules out projective planes of orders 6 and 14 for instance, but does not guarantee the existence of a plane whenn 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,^[27] which in turn implies that there do not exist nine MOLS of order 10.

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

The minimum number of MOLS(n) is known to be 2 for alln except forn = 2 or 6, where it is 1. However, more can be said, namely,^[28]

MacNeish's Theorem: If${\displaystyle n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}}$ is the factorization of the integern into powers of distinct primes${\displaystyle p_1,p_2,\cdots ,p_r}$ then

the minimum number of MOLS(n)${\displaystyle \ge \underset{i}{\min }\{p_i^{{\alpha }_i}-1\}.}$

MacNeish's theorem does not give a very good lower bound, for instance ifn ≡ 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 ifn > 6. On the other hand, it does give the correct value whenn is a power of a prime.

For general composite numbers, the number of MOLS is not known. The first few values starting withn = 2, 3, 4... are 1, 2, 3, 4, 1, 6, 7, 8, ... (sequenceA001438 in theOEIS).

The smallest case for which the exact number of MOLS(n) is not known isn = 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.^[29]

For large enoughn, the number of MOLS is greater than${\displaystyle \sqrt[14.8]{n}}$, thus for everyk, there are only a finite number ofn such that the number of MOLS isk.^[30] Moreover, the minimum is 6 for alln > 90.

A complete set of MOLS(q) exists wheneverq is a prime or prime power. This follows from a construction that is based on afinite fieldGF(q), which only exist ifq is a prime or prime power.^[31] The multiplicative group ofGF(q) is acyclic 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 theq elements ofGF(q) as follows:

α₀ = 0, α₁ = 1, α₂ = λ, α₃ = λ², ..., α_q-1 = λ^q-2.

Now, λ^q-1 = 1 and the product rule in terms of the α's is α_iα_j = α_t, wheret =i +j -1 (modq -1). The Latin squares are constructed as follows, the (i, j)th entry in Latin square L_r (withr ≠ 0) is L_r(i,j) = α_i + α_rα_j, where all the operations occur inGF(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 theintegers modulop, the naming convention above can be dropped and the construction rule can be simplified to L_r(i,j) =i +rj, wherer ≠ 0 andi,j andr are elements ofGF(p) and all operations are inGF(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, aDesarguesian projective plane. There existnon-Desarguesian projective planes and their corresponding complete sets of MOLS can not be obtained from finite fields.^[32]

Anorthogonal array, OA(k,n), of strength two and index one is ann² ×k arrayA (k ≥ 2 andn ≥ 1, integers) with entries from a set of sizen such that within any two columns ofA (strength), every ordered pair of symbols appears in exactly one row ofA (index).^[33]

An OA(s + 2,n) is equivalent tos MOLS(n).^[33]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 labeledr andc denote the row and column of a position in a square and the rest of the row for fixedr andc values is filled with the entry in that position in each of the Latin squares. This process is reversible; given an OA(s,n) withs ≥ 3, choose any two columns to play ther andc 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.

A (geometric) (k,n)-net is a set ofn² elements calledpoints and a set ofkn subsets calledlines orblocks each of sizen with the property that two distinct lines intersect in at most one point. Moreover, the lines can be partitioned intok parallel classes (no two of its lines meet) each containingn lines.^[34]

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

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

To construct a (k + 2,n)-net fromk MOLS(n), represent the MOLS as an orthogonal array, OA(k + 2,n) (seeabove). The ordered pairs of entries in each row of the orthogonal array in the columns labeledr andc, will be considered to be the coordinates of then² points of the net. Each other column (that is, Latin square) will be used to define the lines in a parallel class. Then lines determined by the column labeled L_i will be denoted byl_ij. The points onl_ij will be those with coordinates corresponding to the rows where the entry in the L_i column isj. There are two additional parallel classes, corresponding to ther andc columns. The linesr_j andc_j consist of the points whose first coordinates arej, or second coordinates arej respectively. This construction is reversible.^[35]

For example, the OA(5,4) in the above section can be used to construct a (5,4)-net (an affine plane of order 4). The points on each line are given by (each row below is a parallel class of lines):

l11:	(1,1) (2,2) (3,3) (4,4)	l12:	(1,2) (2,1) (3,4) (4,3)	l13:	(1,3) (2,4) (3,1) (4,2)	l14:	(1,4) (2,3) (3,2) (4,1)
l21:	(1,1) (2,4) (3,2) (4,3)	l22:	(1,2) (2,3) (3,1) (4,4)	l23:	(1,3) (2,2) (3,4) (4,1)	l24:	(1,4) (2,1) (3,3) (4,2)
l31:	(1,1) (2,3) (3,4) (4,2)	l32:	(1,2) (2,4) (3,3) (4,1)	l33:	(1,3) (2,1) (3,2) (4,4)	l34:	(1,4) (2,2) (3,1) (4,3)
r1:	(1,1) (1,2) (1,3) (1,4)	r2:	(2,1) (2,2) (2,3) (2,4)	r3:	(3,1) (3,2) (3,3) (3,4)	r4:	(4,1) (4,2) (4,3) (4,4)
c1:	(1,1) (2,1) (3,1) (4,1)	c2:	(1,2) (2,2) (3,2) (4,2)	c3:	(1,3) (2,3) (3,3) (4,3)	c4:	(1,4) (2,4) (3,4) (4,4)

Atransversal design withk groups of sizen and index λ, denoted T[k, λ;n], is a triple (X, G, B) where:^[36]

• X is a set ofkn varieties;
• G = {G₁,G₂, ...,G_k} is a family ofk n-sets (calledgroups, but not in the algebraic sense) which form a partition ofX;
• B is a family ofk-sets (calledblocks) of varieties such that eachk-set inB intersects each groupG_i in precisely one variety, and any pair of varieties which belong to different groups occur together in precisely λ blocks inB.

The existence of a T[k,1;n] design is equivalent to the existence ofk-2 MOLS(n).^[37]

A transversal design T[k,1;n] is thedualincidence structure of an (k,n)-net. That is, it hasnk points andn² blocks. Each point is inn blocks; each block containsk points. The points fall intok equivalence classes (groups) of sizen so that two points in the same group are not contained in a block while two points in different groups belong to exactly one block.^[38]

For example, using the (5,4)-net of the previous section we can construct a T[5,1;4] transversal design. The block associated with the point (i, j) of the net will be denotedb_ij. The points of the design will be obtained from the following scheme:r_i ↔i,c_j ↔ 5j, andl_ij ↔ 5i +j. The points of the design are thus denoted by the integers 1, ..., 20. The blocks of the design are:

b11:	6 11 16 1 5	b22:	6 13 19 2 10	b33:	6 14 17 3 15	b44:	6 12 18 4 20
b12:	7 12 17 1 10	b21:	7 14 18 2 5	b34:	7 13 16 3 20	b43:	7 11 19 4 15
b13:	8 13 18 1 15	b24:	8 11 17 2 20	b31:	8 12 19 3 5	b42:	8 14 16 4 10
b14:	9 14 19 1 20	b23:	9 12 16 2 15	b32:	9 11 18 3 10	b41:	9 13 17 4 5

The five "groups" are:

6 7 8 9

11 12 13 14

16 17 18 19

1 2 3 4

5 10 15 20

A set ofk MOLS(n) is equivalent to an edge-partition of thecomplete (k + 2)-partite graph K_n,...,n intocomplete subgraphs of orderk + 2.^[10]

Mutually orthogonal Latin squares have a great variety of applications. They are used as a starting point for constructions in the statisticaldesign of experiments,tournament scheduling, anderror correcting and detecting codes. Euler's interest in Graeco-Latin squares arose from his desire to constructmagic squares. The French writerGeorges Perec structured his 1978 novelLife: A User's Manual around a 10×10 Graeco-Latin square.

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC,ISBN 978-1-58488-506-1
• 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,S2CID 82321,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 & Padgett, L.V. (2005).Block Designs: Analysis, Combinatorics and Applications. World Scientific.
• 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
• van Lint, J.H.; Wilson, R.M. (1993),A Course in Combinatorics, Cambridge University Press,ISBN 978-0-521-42260-4

• Leonhard Euler's Puzzle of the 36 Officiers AMS featured column archive (Latin Squares in Practice and Theory II)
• Weisstein, Eric W."36 Officer Problem".MathWorld.
• Euler's work on Latin Squares and Euler Squares at Convergence
• Java Tool which assists in constructing Graeco-Latin squares (it does not construct them by itself) atcut-the-knot
• Anything but square: from magic squares to Sudoku
• Historical facts and correlation with Magic Squares,Javascript Application to solve Graeco-Latin Squares from size 1x1 to 10x10 and relatedsource code (JavaScript in Firefox browser and HTML5 mobile devices)
• Grime, James (8 May 2020)."Euler Squares"(video).YouTube.Brady Haran.Archived from the original on 2021-12-12. Retrieved9 May 2020.

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