For questions on or pertaining to Latin squares.

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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
31 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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0answers
30 views

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
43 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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2answers
35 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
42 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
49 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
37 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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0answers
53 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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0answers
32 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
46 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
55 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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2answers
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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
87 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 ...
2
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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 ...
2
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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 ...
2
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1answer
47 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 ...
2
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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.
0
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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 ...
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2answers
55 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
43 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
114 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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0answers
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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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106 views

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
153 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
41 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
86 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
85 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 ...
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1answer
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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 ...
0
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1answer
73 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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1answer
81 views

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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3answers
150 views

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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2answers
400 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
95 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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2answers
98 views

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 ...
8
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1answer
403 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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1answer
82 views

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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3answers
369 views

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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1answer
94 views

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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1answer
150 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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3answers
169 views

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
134 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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1answer
253 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 ...
2
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1answer
96 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
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2answers
147 views

Latin squares of even order-sub squares

Consider Latin squares of even order that is not of form $2^x$, where every cell is involved in a $2\times 2$ sub square. Here is one such square for order 6: ...
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1answer
71 views

q-ary code/Latin squares

For any value of $q$ the largest number of elements in any q-ary code $C$ of length $4$, distance $3$ is $q^2$. How can we prove that this is attainable iff there are a pair of mutually orthogonal ...
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1answer
137 views

No pandiagonal latin squares with order divisible by 3?

I would like to prove the claim that pandiagonal latin squares, which are of form ...