For questions on or pertaining to Latin squares.

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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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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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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
19 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
35 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
46 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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38 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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50 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
50 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
38 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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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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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
48 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
58 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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101 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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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
38 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
40 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 ...
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1answer
53 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
41 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
68 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
44 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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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 ...
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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
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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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115 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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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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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
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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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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 ...
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How many Sudoku puzzles are there with at least one solution?

A Sudoku puzzle is a 9 by 9 matrix of blanks(which we can represent as 0), and elements of the set {1,2,3,4,5,6,7,8,9}. How many Sudoku puzzles are there with at least one solution. Yes, I am even ...
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1answer
76 views

Probability of occurrence after Latin Hypercube sampling and then random sampling

I am using Latin Hypercube sampling to obtain numbers from a Normally Distributed set of data, so that I get a uniform spread of numbers across the Normal Distribution. I then select a number at ...
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Proof concerning Latin squares

I'm asked to solve this problem : Let $R$ be an $r\times c$ partial Latin rectangle using the numbers $[n]= \{1,2,...,n\}$. Suppose that $r < n$ and $c < n$, and let $N(i)$ be the number of ...
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Find three $10\times10$ orthogonal Latin squares.

Does anyone know if there is a mathematical "trick" in finding mutually orthogonal Latin squares? Or is it basically trial and error?
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411 views

Shift interval of log-normally distributed latin hypercube samples

first of all I'm not sure if this part of StackExchange is the right one because my question is mainly on a way to implement something in MATLAB. Ok, now let me try to pack my whole question in one ...
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1answer
96 views

Partition of a set and Hall's theorem

I have been wrestling with an exercise concerning latin squares from the textbook A First Course in Discrete Mathematics by Ian Anderson. The exercise is formulated thus: A set $S$ of $mn$ ...
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Union notation in Hall's theorem

This is taken from a proof for Hall's theorem based on ideas of R. Rado. I am having trouble with some union notation in the proof. (Substantial parts of the proof is missing here as I am only ...
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407 views

symmetric latin square of order 5

My textbook said if a latin square of order 5 is idempotent and had a 2 in the (1,3) entry, it could not be completed as a symmetric square. But isn't this one such square? Or do I have the ...
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1answer
84 views

Check my proof on Latin Squares

Define a set of $n - 1$ mutually orthogonal squares $A_1, \ldots, A_{n-1}$, where n is a prime number. If we set the element $(i, j)$ in square $A_h$ as $i + hj$ then the squares in the set are ...
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Is there a name for this given type of matrix?

Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$? (The motivation for this ...
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Orthogonal Latin Square 6*6

I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find ...
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proof of the completion of a certain kind of partial latin square

About THE PROOF of the partial latin square has a completion. Let $r, n \in \mathbb N$, with $r \leq n$. Let $P$ be a PLS of order $n$, in which the first $r$ values are used in $n$ entries and the ...
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151 views

Orthogonal Latin Squares help!

I am having trouble with the following question any help would be great! Suppose that $n$ is an odd positive integer with $n\geq3$. Let $A$ be the $n\times n$ Latin square whose rows and columns are ...
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The number of ways of completing this partial Latin square

If we want to fill the empty squares by the numbers $1$, $2$, $3$, $4$, $5$, $6$ so that all the numbers appear in each row and column, how can we find the number of ways to do that? ...
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1answer
139 views

Constructing MOLS

I'm looking for a method to construct a pair of two MOLS of even side 4n, n > 2. Is there a method for this, preferably one that is simple to implement? I believe this book to contain a simple ...
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254 views

Latin Squares - Proving the Unique number of Sudoku that can be generated

I recently read that the Sudokus are just Latin Squares for $n = 9$. I know that proving the number of Latin Squares is considered difficult to generalize in terms of $n$. I would like to know if ...