For questions on or pertaining to Latin squares.

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How to generate centrally symmetric Latin squares?

I'd like to generate centrally symmetric latin squares for arbitrary sizes. For example for size 6 it could be ...
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2answers
160 views

The Hardest Sudoku Puzzle

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers ...
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1answer
61 views

Counting the number of Latin squares

Counting the number of latin squares is a difficult problem. I understand that the common used formula is $n!(n-1)!$ (the number or reduced latin squares). As seen here and in many other pages you can ...
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1answer
49 views

How to count the latin squares of order 4

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. So, Assume that an integer like $4$ is given. How many $...
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107 views

Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
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77 views

Latin squares using fixed word lists

Consider the problem of constructing a latin square of order $N$, using only row and column values from a given word list ($W$) containing some subset of the $N!$ possible word values. For example, ...
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3answers
69 views

Show Latin Square is not a group.

If we fix the first two rows in the above figure, then there are many ways to fill in the remaining rows to obtain a Latin square. Show that none of these Latin squares is the multiplication table of ...
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2answers
41 views

Transversals of Latin Squares

According to this thesis, page $28$, the following Latin Square has $3$ $0$-s transversals: $$\begin{bmatrix}1 & 2 & 3 & 4 & 5\\ 2 & 4 & 1 & 5 & 3\\ 3 & 5 & 4 &...
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1answer
38 views

There exists exactly $p-1$ mututally latin squares for $p=q^d$ where $q$ is a prime number.

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. A pair of latin squares $A=(a_{ij})$ and $B=(b_{ij})$ ...
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1answer
23 views

Proving that a latin square has a set amount of edges.

I was reading this article that states (page 3)that the total number of edges of a latin square when converting it to a graph is $$ n^2 (n-1) $$ Can someone please prove why this is true? Here's ...
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2answers
45 views

Latin square and groups

From what I can gather, table of each group present latin square. But I wonder, how can I give an example of latin square which isn't a result of the operations on group?
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1answer
39 views

Clarification about mutually orthogonal latin squares

This question is related to my previous one but different in its substance. I have a several questions that I am not able to find answers to. My understanding of mutually orthogonal latin squares is ...
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2answers
82 views

Algorithms for mutually orthogonal latin squares - a correct one?

I am very interested in using mutually orthogonal latin squares (MOLS) to reduce the number of test cases but I struggle to find a way how to implement the algorithm. In an article published in a ...
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1answer
59 views

Find combinations of N sets having no more than one item in common.

Problem definition; There are N input sets, of sizes $S_1, S_2, ... ,S_N$. eg; 4 sets - (1a,2a,3a,4a,5a), (1b,2b,3b,4b), (1c,2c,3c), (1d,2d,3d) A combination is made by selecting one item from each ...
4
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1answer
74 views

Room square construction (Howell design)

Using superimposed orthogonal Latin squares to construct a balanced tournament design, I get an n x n array of unique ordered pairs. If I only want unique unordered pairs, I can eliminate half of the ...
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1answer
117 views

Trouble understanding Latin squares and group theory

This is more of a theoretical question, but I'm having trouble understanding why it is that Latin squares are generalizations of a group? I kind of arrived at this question trying to figure out why ...
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52 views

Which graph with an automorphism group isomorphic to the quaternion group $Q_8$ minimizes $|V|+3|E|$?

In Symmetries of partial Latin squares, it is shown that for any graph $\Gamma=(V,E)$ with automorphism group $G$, there is a partial Latin square with $|V|+3|E|+49$ filled cells whose autotopism ...
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Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
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1answer
74 views

Open Questions on Latin Squares and Directed Acyclic Graphs

Every Latin square corresponds to a directed acyclic graph (DAG) with a lattice arrangement, and whose $2N(N-1)$ edges indicate label order (<). For example: ...
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2answers
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Available lists of all latin squares up to order 5?

There are available online lists of the number of all latin squares up to order 11, e.g.: https://oeis.org/A002860. For a permutation-based test of a latin square design, one option is to fit the ...
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1answer
39 views

Rank-reducibility of Latin squares

Consider the following Latin square with rank $N= 9$: $$ \begin{bmatrix} 5 & 3 & 1 & 2 & 4 & 7 & 6 & 8 & 9 \\ 3 & 7 & 9 & 6 & 8 & 4 & 5 &...
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1answer
48 views

polynomial representing a self-orthogonal latin square

I need to show that for $q$, a prime power not equal to 2 or 3, the polynomial $f(x,y) = \lambda x+(1-\lambda)y$ represents a self-orthogonal latin square of order $q$, where $\lambda \in F_{q}$ is ...
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Cycle detection in a pseudo-Latin square

Given a matrix of size $m \times n$ with no repetition of values in rows or columns, is there an efficient method of detecting cycles? For example, here is a sample matrix: 3 5 2 9 7 4 ...
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1answer
57 views

computing in a loop

Consider a loop, i.e. a set $Q$ with operation $\cdot$ such that we have cancellation law and an identity element. For given $x,y\in Q$ consider the equation $$x=((xy)^{-1}_R\cdot z)^{-1}_L$$ where ...
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3answers
82 views

How do I construct two 4 by 4 orthogonal Latin Squares?

I am constructing two, 4 by 4, orthogonal Latin Squares from the alphabet {$a,b,c,d$}. I have already created one Latin Square. Is there a method for constructing the other Latin Square or is it just ...
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2answers
172 views

Balanced Latin Square

For making a good Between-Object user study, this is suggested to use a Latin Square to give all the different conditions, ...
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2answers
60 views

Identify the following table

What is the name of the following table1,2 of two digit numbers? 1 Can be found on pg. 10 of this link. 2 I've left some of the surrounding text for context.
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1answer
71 views

How to show mutually orthogonal latin squares

I have a question concerning mutually orthogonal latin squares (MOLS). Let $ \mathbb F $ be a field of $n\in\mathbb N$ elements. For all $q\in\mathbb F \backslash \{0\}$, define $n\times n $ tables ...
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96 views

Are there Latin squares with no repeats on the diagonal not of the form 2y+x+1(mod n)?

I am looking for a certain kind of latin square (nxn). Rules: No repeats in any column or row (Definition of Latin Square) No repeats in any diagonal including others than the main diagonal. So $\...
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43 views

Latin Squares and Olderogge Code

So I have two Latin Squares, $A$ and $B$ that form a pair of MOLS of order $m$. I then have an Olderogge code formed from $A$ and $B$, where each binary vector of length $m^2$ is encoded as a codeword ...
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2answers
86 views

Question about a symmetric matrix

Let us have a $2n+1\times 2n+1$ symmetric matrix $A$ where $n$ is a nonnegative integer. We write the numbers $1,2,...,2n+1$ in every row and column, in an arbitrary sequence. Prove that in the main ...
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1answer
135 views

Nearest latin square

given a n x n matrix A with integer entries is there any way to find the nearest n x n latin square to it, say, e.g., in the Frobenius norm? I am looking for some type of convex optimization... ...
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124 views

Does every $9 \times 9$ Latin square contain a $3 \times 3$ submatrix containing each symbol in $\{1,2,\ldots,9\}$?

Q: Does every $9 \times 9$ Latin square on the symbol set $\{1,2,\ldots,9\}$ contain a $3 \times 3$ submatrix containing each symbol in $\{1,2,\ldots,9\}$? This one has $1728$ such submatrices, ...
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0answers
102 views

Latin square code dimension

I am need to understand what is the dimension of the code generated by the Olderogge Encoding: Given two mutually orthogonal latin squares, the encoding of a message of $m^2$ bits is: the message ...
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1answer
43 views

Need help understanding a proof about $N_{2}$ latin squares.

I understand what the author is doing in Theorem 1.3.1 intercalate proof, however I don't see how this proof relies on the fact that $n$ needs to be odd. It seems to me the same logic holds for any $...
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1answer
65 views

prove Orthogonal Latin Squares

Suppose that $n$ is an odd positive integer with $n \geq 3$. Let $A$ be the $n \times n$ Latin square whose rows and columns are indexed by the elements of $\mathbb Z_n = \{0, 1, 2, \ldots, n - 1\}$ ...
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1answer
101 views

Constructing orthogonal latin square Parker/Knuth method

I'm working through Knuth; The Art of Computer Programming, Vol. 4 Fascicle 0 and I'm having a little trouble making sense of the method Knuth describes for computing an orthogonal square. The square ...
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1answer
50 views

Can someone clarify the definition for two Latin squares to be orthogonal for me please?

I have been given this definition for two Latin Sqaures to be orthogonal. Let $L, M$ be two $n\times n$ Latin squares with entries taken from the sets $X = \{x_1, x_2,\dots, x_n\}$,$\ \ Y = \{y_1, ...
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1answer
106 views

Orthogonal Latin Square

Find a Latin square orthogonal to the following Latin square: 0 2 1 3 2 0 3 1 3 1 2 0 1 3 0 2 I have done this by using trial and error. But my ...
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1answer
48 views

A Very Elementary Article or Webpage about Secret Sharing

I'm looking for an article or webpage about secret sharing with Latin squares, accessible to middle school students. I searched but found none. Can you help me? Thanks.
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1answer
22 views

Matrix array built from range?

I have the problem on a past exam: Let the array $L$ be defined by: $0\leq i, j \leq n-1$ $$L=[l_{i,j}] \text{where } l_{i.j} \equiv i+j \pmod n$$ Let $n=4$ and write down the array $L=[...
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2answers
67 views

What does a non-mathematician need to google to learn more about latin squares in which each number in each row always has a different successor?

First off, my apologies for the long and convoluted title. I am no mathematician so I don't know the "proper" terms to use... which is exactly my problem: I want to find a/any/one Latin square of ...
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1answer
58 views

Connections between loops (algebraic structure) and graphs

I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this ...
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2answers
127 views

Some kind of latin squares

Consider we have an $n\times n$ square. And for each element $a_{ij}$ there is a $L_{ij}$ set of permissible values(numbers) where $|L_{ij}| = n - 1$. Need to choose a value for each $a_{ij}$ element ...
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Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
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Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
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4answers
275 views

Transforming a latin square into a sudoku

Can any $9\times 9$ - Latin Square be transformed into a sudoku by just exchanging rows and columns (it is allowed to mix row- and column-exchanges arbitarily and there is no limit for the number of ...
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0answers
47 views

Does there exist Latin square critical sets for which deleting any entry results in arbitrarily many completions?

For those familiar with Latin squares terminology, I'll get straight to the point: Q: For all $N \geq 2$, does there exists a critical set $C$ (for a Latin square of any finite order) such that ...
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1answer
105 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
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2answers
117 views

Modified latin square

I'm trying to build a variation of a latin square. In a latin square of size $n$, every row and every column contains a number from $1$ to $n$ exactly once. Given arbitrary $a$ and $b$ such that $n=a\...