Questions tagged [latin-square]

For questions on or pertaining to Latin squares.

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Does $(\Bbb{R}, +)$ admit an irreducible $2$-traversal?

For a given natural number $k$, I'm going to call a subset $T$ of the plane $\Bbb{R}^2$ a $k$-traversal if, for any $x \in \Bbb{R}$, \begin{align*} k &= \operatorname{card} \{(a, b) \in T : a = x\}...
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Are these anti-circulant matrices?

Consider the matrix \begin{pmatrix}1&k+2&2&k+3&\ldots&2k+1&k+1\\k+2&2&k+3&3&\ldots&k+1&1\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&...
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Completion of partial latin squares

I'm reading a PDF about completing partial latin squares. The reference is https://ajc.maths.uq.edu.au/pdf/22/ocr-ajc-v22-p247.pdf There is a fact that doesn't have a proof here, and I can't find it: ...
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Quasigroup from a finite projective plane order 2

I have only seen a quasigroup in terms of a latin square with numbers and so am not sure where a quasigroup comes up in the circumstance of a projective plane. Could someone provide an example of a ...
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1answer
51 views

Combinatorics: Existence of a Latin Square

Let $k_n$ be the smallest number such that given an n by n grid with $k_n$ arbitrary numbers in the top left corner and n arbitrary numbers in every other cell, a number can be chosen from each cell ...
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1answer
66 views

Finding intercalates within a (reduced) latin square

I need confirmation on if my intuition on finding intercalates is correct, suppose we have the following reduced latin square \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 &...
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Reduced latin square that does not fit quadrangle criterion

I have been having trouble figuring out exactly visually when a certain reduced Latin square does not fit the quadrangle criterion; especially when you can flip things to match. So this is my attempt ...
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32 views

Multiplication table of a commutative quasigroup is a symmetric latin square. Is the converse also true? [closed]

Or can we find an example where the Latin square is symmetric but its corresponding quasi group is not commutative?
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55 views

Existence of Latin Squares without symmetry around main-diagonal

A Latin square of order $n$ is a matrix $L$ with entries from $[n] \equiv \{0, \dots, n-1\}$ such that each row and column contains every symbol from $[n]$ once. For which orders does there exist a ...
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34 views

How many magmas exist on $n$-element set

It is clear that we can make $n^{n^2}$ Latin squares (I think that this is no real Latin square, but I don't know how to name it) for $n$-element set, but I have heard that some magmas will be ...
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Subsets of Reduced Latin Squares of a Given Order

I would like to know conventional methods for specifying subsets of reduced latin squares of a given order that have certain properties under multiplication. #1) It seems redundant to indicate that ...
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Application of Latin Squares

There are 36 officers, six officers of six different ranks in each of 6 regiments. Find an arrangement of the 36 officers in a $6\times 6$ square formation such that each row and each column contains ...
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Generating and Referring to Reduced Latin Squares

I've been looking online for information about latin squares and I came across this overview: http://ccom.uprrp.edu/~labemmy/Wordpress/wp-content/uploads/2010/11/4_Presentation_Some-Properties-of-...
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How Do I Refer to Each Latin Square of a Given Order?

My understanding is that there are four normalized latin squares of order four. What is the preferred way to refer to each of those four? Do they each have a different number, letter, name, etc.? Is ...
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Designations of Reduced Latin Squares

I suspect that each reduced latin square of order n $> 3$ has a conventional designation (a name, a number, etc.) but I have yet to find where they are listed. I have found many representations of ...
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For which integers $k$ there exists an $(k-1)$-$(k, k, 1)$ orthogonal array?

This is what I've found out so far: Since the first $k-1$ columns must have all the possible $k^{k-1}$ strings of length $k-1$ in exactly one row and the rows can be reorganized without changing the "...
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1answer
30 views

Solutions to a Graeco-Latin grid?

I have a 4x4 grid of 16 different elements. Each element is composed of two numbers that can be 1-4. I want to know if I can find a permutation of the grid in which every row and column has 1-4 for ...
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Quadrangle criterion for a matrix

I need help understanding the quadrangle criterion. First of all, I find it very hard to find anything related to it. The only two things I came up with are these: "Reconstruction of Multiplication ...
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1answer
48 views

Would an invalid Sudoku puzzle that becomes valid when you assume its validity be valid?

Suppose you have a sudoku puzzle that you will want to solve using logic. Furthermore suppose you solve the puzzle until you reach a point where a single cell can have two possible values ($a$ or $b$ ...
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1answer
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A latin square of order $n$ has a weak transversal.

A matrix of order $n$ with no repeated element in a row or in a column has a weak transversal. In particular, a latin square of order $n$ has a weak transversal. Proof. We prove the theorem by ...
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36 views

Minimal number of clues required to uniquely determine a Latin Square

Let's suppose I want to fill a n*n array with numbers occuring from 1 to n. One number is allowed to appear only once in every column/row. The solution becomes unique when there is only one possible ...
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1answer
23 views

Cyclic Shift of Latin Squares

I'm trying to solve this following problem on Latin squares: "Suppose that the first row of an $n \times n$ array is \begin{align*} x_1 \ \ x_2 \ \ x_3 \ldots x_{n-1} \ \ x_n, \end{align*} and ...
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How-to leverage two unique orderings for a list to produce a single unique ordering for that list?

The goal: Two people agree on an arbitrary ordering of a list of times. They pre-select some information. Then they reveal just enough information to uncover one item at a time. e.g.: Bob selects "...
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What is a Latin Square?

I have recently seen a problem that used the term "latin square". I was wondering, what is a latin square? I know that it is actual square. Is a latin square a square like a Sudoku puzzle or is it ...
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How many ways to colour a $4 \times 4$ grid using four colours subject to three constraints

In how many ways can a $4 \times 4$ square grid be coloured using four different colours so that no colour is repeated in any row, column, or along the two main diagonals. For clarity, one valid ...
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1answer
155 views

Latin square design for a social “mixer”

I'm looking for the name of the combinatorial satisfying the following requirements: There are $n$ different tasks. There are $n$ different participants. There are $n$ phases of the experiment ...
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1answer
109 views

Orthogonal $4x4$ Latin Squares

Suppose that two orthogonal $4x4$ Latin Squares both have $1,2,3,4$ as the main diagonal. Is it possible for both of them to have the same $(2,3)$ entry? My thinking was to write out Latin squares of ...
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Question from the British Mathematical Olympiad [duplicate]

Question: A $5×5$ square is divided into 25 unit squares. One of the numbers $1,2,3,4,5$ is inserted into each of the unit squares in such away that each row, each column and each of the two ...
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2answers
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Let $L$ be a latin square whose entries are $L_{ij} = i+j (mod n)$. Then there are no $n$ distinct entries from unique rows and columns of $L$

Suppose $n$ is even. Consider an $n$ x $n$ Latin square L, defined by $L_{ij} = i+j (mod n)$ I want to show that if I choose $n$ values from distinct rows and columns, then at least two of the chosen ...
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1answer
76 views

Problem 2.2 from Jukna's “Extremal Combinatorics”

This is problem 2.2 from Junka's Extremal Combinatorics. The problem is as follows: Let $A=(a_{ij})$ be an $n \times n$ matrix with $n \geq 4$. The matrix is filled with integers, and each integer ...
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Find two different latin squares of order $5$

In the following question I am trying to find two different Latin squares of order $5$ Latin Square #1 \begin{array} & &1 &2 &3 &4 &5 \\ &5 &1 &2 &3 &4 \...
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1answer
58 views

Perfect codes over non-prime-power alphabets and Latin squares.

I'm reading through Peter J. Cameron's Combinatorics: Topics, Techniques, Algorithms, specifically the section on perfect e-error-correcting codes over alphabets on non-prime-power length. Consider ...
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1answer
149 views

How to calculate the direct product of Latin squares

In the example below a 2x2 Latin square is multiplied by a 3x3 Latin square which gives a 6x6 Latin square. My question is what it the method for multiplying these two different sized Latin squares ...
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1answer
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Cycles in $5×5$ Latin squares (solution found)

One of my favorite puzzle formats is KenKen, in which you are to find an $n×n$ Latin square to be filled with whole numbers from 1 to $n$, given sums, differences, products or quotients of various ...
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Randomising with adjacency constraints

I am a high school political science teacher looking for some help in implementing a classroom exercise. It's a puzzle of sorts, I suppose. I've tried Googling the issue for a while, but my GoogleFu ...
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271 views

Is there a sudoku (Latin Square Pattern) state in a Rubik's cube $6\times6\times6?$

Suppose, Initial state Rubik's Cube 6x6x6 444444 444444 444444 444444 444444 444444 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 ...
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Which numbers satisfy conditions for these kind of latin squares?

This is a question: Prove that there are infinite natural numbers such that there is a latin square with size n & on the main & subsidiary diagonal numbers {1,2,...,n} appear. I want all ...
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1answer
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Can a solved Sudoku game have an invalid region if all rows and columns are valid? [closed]

Given a $9 \times 9$ solved Sudoku game with $3 \times 3$ regions, is it possible that one (or more) of the regions are invalid if all rows and columns are valid (i.e. have a unique sequence of $1-9$)?...
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111 views

What's the difference between the latin square and the matrix? [closed]

What's the difference between the latin square and the matrix?
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Formula for MOLS generation when n is prime power - really?

I remember from the university that there is a procedure to simply generate MOLS when n is a prime number. Basically it is about cyclic rotation of the symbols in the rows, for square k by k positions....
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129 views

How Latin squares remove variation in experiment design? (2-way blocking)

I read that latin squares are used to design of experiments when there are 2 sources of nuisance (factors). In that setting, every treatment appears exactly once in each row (factor 1) and column (...
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1answer
92 views

Symmetric latin square of order 9 & 10 ? (focusing the diagonal)

The smallest possible symmetric latin square is the order of 4 which is $$\matrix{1&2&3&4\cr 2&1&4&3\cr 3&4&1&2\cr 4&3&2&1}$$ but i'm also wondering if ...
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425 views

When is a Sudoku like table solvable

Given a $n\times n$ table is it possible to fill each cell with one of the numbers $1,2,3,\cdots,n$ such that in each column,each row and each diagonal (i.e Denoting $(x,y)$ as number of column and ...
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Can equi-$n$-squares be seen as wavy-latin $n$-squares?

An equi-$n$-square (S.K.Stein, 1975) is a square $n$-matrix with digits (colors) $0 \ldots n-1$ occurring $n$ times each. Latin squares have the stronger property that each digit occurs once in each ...
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Proving the orthogonality of Latin squares

I know there are already several posts on how to prove the orthogonality of Latin squares that may be using a better proof, however there is a particular part in a proof I read that I could use some ...
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Weakly non-completable partial Latin square

An empty cell in a partial Latin square (pLs) is said to be forced if it has a unique admissible entry (compatible with the definition of a Latin square). Attempting to complete a given pLs, one can ...
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184 views

What is the operation of quasigroup in this example?

This example is taken from here. A quasigroup of order n is a pair (Q, o), where Q is a set of size n and "o" is a binary operation on Q s. t. for every pair of elements $a,b\in Q$ the equations $a$ ...
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71 views

Minimum critical set size for Sudoku Pairs

Define $Sudoku$ $Pair$ as a pair of mutually orthogonal Sudoku Squares. For example: $$ \begin{bmatrix} 55 & 18 & 66 & | & 71 & 89 & 32 & | & 93 & 24 & 27 \\ ...
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117 views

One-to-one correspondence between n-edge-colourings of $K_{n,n}$ and Latin squares

I need help with this. I don't know how to do this (especially case b) ). a) An $n \times n$ array $A = (a_{ij})$, whose entries are taken from some set $S$ of $n$ symbols, is called a Latin square ...
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How many structurally distinct Latin squares are there of order 7?

What is the 7th term of https://oeis.org/A264603? A264603 Number of structurally distinct Latin squares of order n. "Structurally distinct" means that the squares cannot be made identical by means ...