In combinatorics and in experimental design, a **Latin square** is an *n* × *n* array filled with *n* different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin square is

- History
- Reduced form
- Properties
- Orthogonal array representation
- Equivalence classes of Latin squares
- Number
- Examples
- Transversals and rainbow matchings
- Algorithms
- Applications
- Statistics and mathematics
- Error correcting codes
- Mathematical puzzles
- Board games
- Agronomic research
- Heraldry
- Generalizations
- See also
- Notes
- References
- Further reading
- External links

A | B | C |

C | A | B |

B | C | A |

The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols,^{ [1] } but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3. Euler began the general theory of Latin squares.

The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years.^{ [2] }

A Latin square is said to be *reduced* (also, *normalized* or *in standard form*) if both its first row and its first column are in their natural order.^{ [3] } For example, the Latin square above is not reduced because its first column is A, C, B rather than A, B, C.

Any Latin square can be reduced by permuting (that is, reordering) the rows and columns. Here switching the above matrix's second and third rows yields the following square:

A | B | C |

B | C | A |

C | A | B |

This Latin square is reduced; both its first row and its first column are alphabetically ordered A, B, C.

If each entry of an *n*×*n* Latin square is written as a triple (*r*,*c*,*s*), where *r* is the row, *c* is the column, and *s* is the symbol, we obtain a set of *n*^{2} triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the Latin square

1 | 2 | 3 |

2 | 3 | 1 |

3 | 1 | 2 |

is

- { (1, 1, 1), (1, 2, 2), (1, 3, 3), (2, 1, 2), (2, 2, 3), (2, 3, 1), (3, 1, 3), (3, 2, 1), (3, 3, 2) },

where for example the triple (2, 3, 1) means that in row 2 and column 3 there is the symbol 1. Orthogonal arrays are usually written in array form where the triples are the rows, such as:

r | c | s |
---|---|---|

1 | 1 | 1 |

1 | 2 | 2 |

1 | 3 | 3 |

2 | 1 | 2 |

2 | 2 | 3 |

2 | 3 | 1 |

3 | 1 | 3 |

3 | 2 | 1 |

3 | 3 | 2 |

The definition of a Latin square can be written in terms of orthogonal arrays:

- A Latin square is a set of
*n*^{2}triples (*r*,*c*,*s*), where 1 ≤*r*,*c*,*s*≤*n*, such that all ordered pairs (*r*,*c*) are distinct, all ordered pairs (*r*,*s*) are distinct, and all ordered pairs (*c*,*s*) are distinct.

This means that the *n*^{2} ordered pairs (*r*, *c*) are all the pairs (*i*, *j*) with 1 ≤ *i*, *j* ≤ *n*, once each. The same is true of the ordered pairs (*r*, *s*) and the ordered pairs (*c*, *s*).

The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.

Many operations on a Latin square produce another Latin square (for example, turning it upside down).

If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be * isotopic * to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called *isotopy classes*, such that two squares in the same class are isotopic and two squares in different classes are not isotopic.

Another type of operation is easiest to explain using the orthogonal array representation of the Latin square. If we systematically and consistently reorder the three items in each triple (that is, permute the three columns in the array form), another orthogonal array (and, thus, another Latin square) is obtained. For example, we can replace each triple (*r*,*c*,*s*) by (*c*,*r*,*s*) which corresponds to transposing the square (reflecting about its main diagonal), or we could replace each triple (*r*,*c*,*s*) by (*c*,*s*,*r*), which is a more complicated operation. Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also parastrophes) of the original square.^{ [4] }

Finally, we can combine these two equivalence operations: two Latin squares are said to be *paratopic*, also *main class isotopic*, if one of them is isotopic to a conjugate of the other. This is again an equivalence relation, with the equivalence classes called *main classes*, *species*, or *paratopy classes*.^{ [4] } Each main class contains up to six isotopy classes.

There is no known easily computable formula for the number *L _{n}* of

A simple and explicit formula for the number of Latin squares was published in 1992, but it is still not easily computable due to the exponential increase in the number of terms. This formula for the number *L _{n}* of

where *B*_{n} is the set of all *n* × *n* {0, 1} matrices, σ_{0}(*A*) is the number of zero entries in matrix A, and per(*A*) is the permanent of matrix A.^{ [6] }

The table below contains all known exact values. It can be seen that the numbers grow exceedingly quickly. For each n, the number of Latin squares altogether (sequence A002860 in the OEIS ) is *n*! (*n*-1)! times the number of reduced Latin squares (sequence A000315 in the OEIS ).

n | reduced Latin squares of size n (sequence A000315 in the OEIS ) | all Latin squares of size n (sequence A002860 in the OEIS ) |
---|---|---|

1 | 1 | 1 |

2 | 1 | 2 |

3 | 1 | 12 |

4 | 4 | 576 |

5 | 56 | 161,280 |

6 | 9,408 | 812,851,200 |

7 | 16,942,080 | 61,479,419,904,000 |

8 | 535,281,401,856 | 108,776,032,459,082,956,800 |

9 | 377,597,570,964,258,816 | 5,524,751,496,156,892,842,531,225,600 |

10 | 7,580,721,483,160,132,811,489,280 | 9,982,437,658,213,039,871,725,064,756,920,320,000 |

11 | 5,363,937,773,277,371,298,119,673,540,771,840 | 776,966,836,171,770,144,107,444,346,734,230,682,311,065,600,000 |

12 | 1.62 × 10^{44} | |

13 | 2.51 × 10^{56} | |

14 | 2.33 × 10^{70} | |

15 | 1.50 × 10^{86} |

For each n, each isotopy class (sequence A040082 in the OEIS ) contains up to (*n*!)^{3} Latin squares (the exact number varies), while each main class (sequence A003090 in the OEIS ) contains either 1, 2, 3 or 6 isotopy classes.

n | main classes | isotopy classes | structurally distinct squares |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 1 | 1 | 1 |

3 | 1 | 1 | 1 |

4 | 2 | 2 | 12 |

5 | 2 | 2 | 192 |

6 | 12 | 22 | 145,164 |

7 | 147 | 564 | 1,524,901,344 |

8 | 283,657 | 1,676,267 | |

9 | 19,270,853,541 | 115,618,721,533 | |

10 | 34,817,397,894,749,939 | 208,904,371,354,363,006 | |

11 | 2,036,029,552,582,883,134,196,099 | 12,216,177,315,369,229,261,482,540 |

The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflexion, and/or permutation of the symbols) for n = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively (sequence A264603 in the OEIS ) .

We give one example of a Latin square from each main class up to order five.

They present, respectively, the multiplication tables of the following groups:

- {0} – the trivial 1-element group
- – the binary group
- – cyclic group of order 3
- – the Klein four-group
- – cyclic group of order 4
- – cyclic group of order 5
- the last one is an example of a quasigroup, or rather a loop, which is not associative.

A **transversal** in a Latin square is a choice of *n* cells, where each row contains one cell, each column contains one cell, and there is one cell containing each symbol.

One can consider a Latin square as a complete bipartite graph in which the rows are vertices of one part, the columns are vertices of the other part, each cell is an edge (between its row and its column), and the symbols are colors. The rules of the Latin squares imply that this is a proper edge coloring. With this definition, a Latin transversal is a matching in which each edge has a different color; such a matching is called a rainbow matching.

Therefore, many results on Latin squares/rectangles are contained in papers with the term "rainbow matching" in their title, and vice versa.^{ [7] }

Some Latin squares have no transversal. For example, when *n* is even, an *n*-by-*n* Latin square in which the value of cell *i,j* is (*i*+*j*) mod *n* has no transversal. Here are two examples:

In 1967, H. J. Ryser conjectured that, when *n* is **odd**, every *n*-by-*n* Latin square has a transversal.^{ [8] }

In 1975, S. K. Stein and Brualdi conjectured that, when *n* is **even**, every *n*-by-*n* Latin square has a **partial** transversal of size *n*-1.^{ [9] }

A more general conjecture of Stein is that a transversal of size *n*-1 exists not only in Latin squares but also in any *n*-by-*n* array of *n* symbols, as long as each symbol appears exactly *n* times.^{ [8] }

Some weaker versions of these conjectures have been proved:

- Every
*n*-by-*n*Latin square has a partial transversal of size 2*n*/3.^{ [10] } - Every
*n*-by-*n*Latin square has a partial transversal of size*n -*sqrt(*n*).^{ [11] } - Every
*n*-by-*n*Latin square has a partial transversal of size*n -*11 log_{2}^{2}(*n*).^{ [12] }

For small squares it is possible to generate permutations and test whether the Latin square property is met. For larger squares, Jacobson and Matthews' algorithm allows sampling from a uniform distribution over the space of *n* × *n* Latin squares.^{ [13] }

- In the design of experiments, Latin squares are a special case of
*row-column designs*for two blocking factors:^{ [14] }^{ [15] }

- In algebra, Latin squares are related to generalizations of groups; in particular, Latin squares are characterized as being the multiplication tables (Cayley tables) of quasigroups. A binary operation whose table of values forms a Latin square is said to obey the Latin square property.

Sets of Latin squares that are orthogonal to each other have found an application as error correcting codes in situations where communication is disturbed by more types of noise than simple white noise, such as when attempting to transmit broadband Internet over powerlines.^{ [16] }^{ [17] }^{ [18] }

Firstly, the message is sent by using several frequencies, or channels, a common method that makes the signal less vulnerable to noise at any one specific frequency. A letter in the message to be sent is encoded by sending a series of signals at different frequencies at successive time intervals. In the example below, the letters A to L are encoded by sending signals at four different frequencies, in four time slots. The letter C, for instance, is encoded by first sending at frequency 3, then 4, 1 and 2.

The encoding of the twelve letters are formed from three Latin squares that are orthogonal to each other. Now imagine that there's added noise in channels 1 and 2 during the whole transmission. The letter A would then be picked up as:

In other words, in the first slot we receive signals from both frequency 1 and frequency 2; while the third slot has signals from frequencies 1, 2 and 3. Because of the noise, we can no longer tell if the first two slots were 1,1 or 1,2 or 2,1 or 2,2. But the 1,2 case is the only one that yields a sequence matching a letter in the above table, the letter A. Similarly, we may imagine a burst of static over all frequencies in the third slot:

Again, we are able to infer from the table of encodings that it must have been the letter A being transmitted. The number of errors this code can spot is one less than the number of time slots. It has also been proven that if the number of frequencies is a prime or a power of a prime, the orthogonal Latin squares produce error detecting codes that are as efficient as possible.

The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.^{ [19] }

The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that nine particular 3×3 adjacent subsquares must also contain the digits 1–9 (in the standard version). See also Mathematics of Sudoku.

The more recent KenKen puzzles are also examples of Latin squares.

Latin squares have been used as the basis for several board games, notably the popular abstract strategy game Kamisado.

Latin squares are used in the design of agronomic research experiments to minimise experimental errors.^{ [20] }

The Latin square also figures in the arms of the Statistical Society of Canada,^{ [21] } being specifically mentioned in its blazon. Also, it appears in the logo of the International Biometric Society.^{ [22] }

- A Latin rectangle is a generalization of a Latin square in which there are
*n*columns and*n*possible values, but the number of rows may be smaller than*n*. Each value still appears at most once in each row and column. - A Graeco-Latin square is a pair of two Latin squares such that, when one is laid on top of the other, each ordered pair of symbols appears exactly once.
- A Latin hypercube is a generalization of a Latin square from two dimensions to multiple dimensions.

- Block design
- Combinatorial design
- Eight queens puzzle
- Futoshiki
- Magic square
- Problems in Latin squares
- Rook's graph, a graph that has Latin squares as its colorings
- Sator Square
- Vedic square
- Word square

- ↑ Wallis, W. D.; George, J. C. (2011),
*Introduction to Combinatorics*, CRC Press, p. 212, ISBN 978-1-4398-0623-4 - ↑ Colbourn, Charles J.; Dinitz, Jeffrey H. (2 November 2006).
*Handbook of Combinatorial Designs*(2nd ed.). CRC Press. p. 12. ISBN 9781420010541 . Retrieved 28 March 2017. - ↑ Dénes & Keedwell 1974 , p. 128
- 1 2 Dénes & Keedwell 1974 , p. 126
- ↑ van Lint & Wilson 1992 , pp. 161-162
- ↑ Jia-yu Shao; Wan-di Wei (1992). "A formula for the number of Latin squares".
*Discrete Mathematics*.**110**(1–3): 293–296. doi:10.1016/0012-365x(92)90722-r. - ↑ Gyarfas, Andras; Sarkozy, Gabor N. (2012). "Rainbow matchings and partial transversals of Latin squares". arXiv: 1208.5670 [CO math. CO].
- 1 2 Aharoni, Ron; Berger, Eli; Kotlar, Dani; Ziv, Ran (2017-01-04). "On a conjecture of Stein".
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*.**87**(2): 203–211. doi:10.1007/s12188-016-0160-3. ISSN 0025-5858. S2CID 119139740. - ↑ Stein, Sherman (1975-08-01). "Transversals of Latin squares and their generalizations".
*Pacific Journal of Mathematics*.**59**(2): 567–575. doi: 10.2140/pjm.1975.59.567 . ISSN 0030-8730. - ↑ Koksma, Klaas K. (1969-07-01). "A lower bound for the order of a partial transversal in a latin square".
*Journal of Combinatorial Theory*.**7**(1): 94–95. doi: 10.1016/s0021-9800(69)80009-8 . ISSN 0021-9800. - ↑ Woolbright, David E (1978-03-01). "An n × n Latin square has a transversal with at least n−n distinct symbols".
*Journal of Combinatorial Theory, Series A*.**24**(2): 235–237. doi:10.1016/0097-3165(78)90009-2. ISSN 0097-3165. - ↑ Hatami, Pooya; Shor, Peter W. (2008-10-01). "A lower bound for the length of a partial transversal in a Latin square".
*Journal of Combinatorial Theory, Series A*.**115**(7): 1103–1113. doi:10.1016/j.jcta.2008.01.002. ISSN 0097-3165. - ↑ Jacobson, M. T.; Matthews, P. (1996). "Generating uniformly distributed random latin squares".
*Journal of Combinatorial Designs*.**4**(6): 405–437. doi:10.1002/(sici)1520-6610(1996)4:6<405::aid-jcd3>3.0.co;2-j. - ↑ Bailey, R.A. (2008), "6 Row-Column designs and 9 More about Latin squares",
*Design of Comparative Experiments*, Cambridge University Press, ISBN 978-0-521-68357-9, MR 2422352 - ↑ Shah, Kirti R.; Sinha, Bikas K. (1989), "4 Row-Column Designs",
*Theory of Optimal Designs*, Lecture Notes in Statistics,**54**, Springer-Verlag, pp. 66–84, ISBN 0-387-96991-8, MR 1016151 - ↑ Colbourn, C.J.; Kløve, T.; Ling, A.C.H. (2004). "Permutation arrays for powerline communication".
*IEEE Trans. Inf. Theory*.**50**: 1289–1291. doi:10.1109/tit.2004.828150. S2CID 15920471. - ↑
*Euler's revolution*, New Scientist, 24 March 2007, pp 48–51 - ↑ Huczynska, Sophie (2006). "Powerline communication and the 36 officers problem".
*Philosophical Transactions of the Royal Society A*.**364**(1849): 3199–3214. doi:10.1098/rsta.2006.1885. PMID 17090455. S2CID 17662664. - ↑ C. Colbourn (1984). "The complexity of completing partial latin squares".
*Discrete Applied Mathematics*.**8**: 25–30. doi:10.1016/0166-218X(84)90075-1. - ↑ http://joas.agrif.bg.ac.rs/archive/article/59 | The application of Latin square in agronomic research
- ↑ "Letters Patent Confering the SSC Arms".
*ssc.ca*. Archived from the original on 2013-05-21. - ↑ The International Biometric Society Archived 2005-05-07 at the Wayback Machine

In linear algebra, the **determinant** is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix *A* is denoted det(*A*), det *A*, or |*A*|.

In linear algebra, the **rank** of a matrix A is the dimension of the vector space generated by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.

In linear algebra, the **column space** of a matrix *A* is the span of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.

In linear algebra, an **orthogonal matrix**, or **orthonormal matrix**, is a real square matrix whose columns and rows are orthonormal vectors.

In mathematics, a **square matrix** is a matrix with the same number of rows and columns. An *n*-by-*n* matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied.

In linear algebra, the **transpose** of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix **A** by producing another matrix, often denoted by **A**^{T}.

In mathematics, especially in applications of linear algebra to physics, the **Einstein notation** or **Einstein summation convention** is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

In mathematics, particularly in matrix theory, a **permutation matrix** is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows or columns of the matrix A.

In linear algebra, a square matrix is called **diagonalizable** or **non-defective** if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . For a finite-dimensional vector space , a linear map is called **diagonalizable** if there exists an ordered basis of consisting of eigenvectors of . These definitions are equivalent: if has a matrix representation as above, then the column vectors of form a basis of eigenvectors of , and the diagonal entries of are the corresponding eigenvalues of ; with respect to this eigenvector basis, is represented by . **Diagonalization** is the process of finding the above and .

In combinatorics, two Latin squares of the same size (*order*) are said to be *orthogonal* 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 of **mutually orthogonal Latin squares**. This concept of orthogonality in combinatorics is strongly related to the concept of blocking 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.

In mathematics, a **Hadamard matrix**, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The *n*-dimensional parallelotope spanned by the rows of an *n*×*n* Hadamard matrix has the maximum possible *n*-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem.

In mathematics, the **kernel** of a linear map, also known as the **null space** or **nullspace**, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map *L* : *V* → *W* between two vector spaces V and W, the kernel of L is the vector space of all elements **v** of V such that *L*(**v**) = **0**, where **0** denotes the zero vector in W, or more symbolically:

**Combinatorial design theory** is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of *balance* and/or *symmetry*. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.

**Latin squares** and **quasigroups** are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of some very small *orders*, which is the side length of the square, or the number of elements in the equivalent quasigroup.

In mathematics, **Bondy's theorem** is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972.

In mathematics, an **orthostochastic matrix** is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix.

In mathematics, an **orthogonal array** is a "table" (array) whose entries come from a fixed finite set of symbols, arranged in such a way that there is an integer *t* so that for every selection of *t* columns of the table, all ordered *t*-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number *t* is called the *strength* of the orthogonal array. Here is a simple example of an orthogonal array with symbol set {1,2} and strength 2:

In mathematics, a **matrix** is a rectangular *array* or *table* of numbers, symbols, or expressions, arranged in *rows* and *columns*. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:

In combinatorial mathematics, a **Latin rectangle** is an *r* × *n* matrix, using n symbols, usually the numbers 1, 2, 3, ..., *n* or 0, 1, ..., *n* − 1 as its entries, with no number occurring more than once in any row or column.

**Hadamard's maximal determinant problem**, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size *n* is 2^{n−1} times the maximal determinant of a {0,1} matrix of size *n*−1. The problem was posed by Hadamard in the 1893 paper in which he presented his famous determinant bound and remains unsolved for matrices of general size. Hadamard's bound implies that {1, −1}-matrices of size *n* have determinant at most *n*^{n/2}. Hadamard observed that a construction of Sylvester produces examples of matrices that attain the bound when *n* is a power of 2, and produced examples of his own of sizes 12 and 20. He also showed that the bound is only attainable when *n* is equal to 1, 2, or a multiple of 4. Additional examples were later constructed by Scarpis and Paley and subsequently by many other authors. Such matrices are now known as Hadamard matrices. They have received intensive study.

- Bailey, R.A. (2008). "6 Row-Column designs and 9 More about Latin squares".
*Design of Comparative Experiments*. Cambridge University Press. ISBN 978-0-521-68357-9. MR 2422352. - 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. - Shah, Kirti R.; Sinha, Bikas K. (1989). "4 Row-Column Designs".
*Theory of Optimal Designs*. Lecture Notes in Statistics.**54**. Springer-Verlag. pp. 66–84. ISBN 0-387-96991-8. MR 1016151. - van Lint, J. H.; Wilson, R. M. (1992).
*A Course in Combinatorics*. Cambridge University Press. p. 157. ISBN 0-521-42260-4.

- Dénes, J. H.; Keedwell, A. D. (1991).
*Latin squares: New developments in the theory and applications*. Annals of Discrete Mathematics.**46**. Paul Erdős (foreword). Amsterdam: Academic Press. ISBN 0-444-88899-3. MR 1096296. - Hinkelmann, Klaus; Kempthorne, Oscar (2008).
*Design and Analysis of Experiments*. I, II (Second ed.). Wiley. ISBN 978-0-470-38551-7. MR 2363107.- Hinkelmann, Klaus; Kempthorne, Oscar (2008).
*Design and Analysis of Experiments, Volume I: Introduction to Experimental Design*(Second ed.). Wiley. ISBN 978-0-471-72756-9. MR 2363107. - Hinkelmann, Klaus; Kempthorne, Oscar (2005).
*Design and Analysis of Experiments, Volume 2: Advanced Experimental Design*(First ed.). Wiley. ISBN 978-0-471-55177-5. MR 2129060.

- Hinkelmann, Klaus; Kempthorne, Oscar (2008).
- Knuth, Donald (2011).
*The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1*. Reading, Massachusetts: Addison-Wesley. ISBN 978-0-201-03804-0. - Laywine, Charles F.; Mullen, Gary L. (1998).
*Discrete mathematics using Latin squares*. Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: John Wiley & Sons, Inc. ISBN 0-471-24064-8. MR 1644242. - Shah, K. R.; Sinha, Bikas K. (1996). "Row-column designs". In S. Ghosh and C. R. Rao (ed.).
*Design and analysis of experiments*. Handbook of Statistics.**13**. Amsterdam: North-Holland Publishing Co. pp. 903–937. ISBN 0-444-82061-2. MR 1492586. - Raghavarao, Damaraju (1988).
*Constructions and Combinatorial Problems in Design of Experiments*(corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0-486-65685-3. MR 1102899. - Street, Anne Penfold; Street, Deborah J. (1987).
*Combinatorics of Experimental Design*. New York: Oxford University Press. ISBN 0-19-853256-3. MR 0908490. - Berger, Paul D.; Maurer, Robert E.; Celli, Giovana B. (November 28, 2017).
*Experimental Design with Applications in Management, Engineering, and the Sciences*(2nd edition (November 28, 2017) ed.). Springer. pp. 267–282.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.