For questions on or pertaining to Latin squares.

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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
44 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
55 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
90 views

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
46 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
60 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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110 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
196 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
71 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
85 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 ...
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1answer
246 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
70 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
75 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
202 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
72 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
114 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
150 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? ...
3
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1answer
92 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
187 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 ...
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1answer
85 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
136 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
64 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
124 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 ...
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1answer
68 views

Could you show me an example of an order 7 pandiagonal latin square?

Could you show me an example of an order 7 pandiagonal latin square? A pandiagonal latin square is one where no broken diagonal contains repeated symbols. I have found examples for smaller order but ...
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87 views

Can we find two mutually orthogonal diagonal latin squares of orders $4$ and $8$?

Can we find two mutually orthogonal diagonal latin squares of orders $4$ and $8$? A diagonal Latin square is a Latin square of order $n$ where the symbols $1$ thru $n$ fil both the forward diagonal ...
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1answer
311 views

Turning affine planes into projective planes

How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$? Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point ...
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1answer
182 views

Latin squares that don't come from group tables?

For what values on $n$ can we find an order $n$ Latin square that does not come from a group table? I understand that most Latin squares can be constructed from group tables...
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179 views

A pair of MOLS of order 15?

Is there any place where I can find a pair of MOLS(mutually orthogonal latin squares) of order 15? I can't seem to find a place where it's spelled out explicitly.
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1answer
49 views

Directory of MOLS that are prime powers?

I had stumbled across a page earlier today that contained the maximally sized sets for mutually orthogonal latin squares of prime power order. I cannot, however, find this page again. Would you be ...
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1answer
107 views

Cayley tables- for any order $n$ there is some latin square with no orthogonal pair

How can we prove, using Cayley tables, that for any order $n$ there is some latin square with no orthogonal pair? What would this mean for the Cayley table regarding transversals? I am not entirely ...
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2answers
120 views

Mutually orthogonal latin squares of order $mn$ from order $m$ and order $n$

Given a pair of mutually orthogonal latin squares (MOLS) of order $m$ and a pair of MOLS of order $n$, how would we construct a pair of MOLS of order $mn$? [EDIT: MOLS means Mutually Orthogonal Latin ...
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1answer
134 views

$6\times 6$ partial mutually orthogonal latin squares

I know that $6\times 6$ MOLS cannot be constructed, but if I am not mistaken we can draw up two MOLS that are $6 \times 6$ with $34$ distinct pairs of symbols. However, I am not able to find this ...
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1answer
160 views

Proof for latin squares- disjoint transversals

I cannot find a proof for the following theorem anywhere: A latin square has an orthogonal mate iff it can be decomposed into disjoint transversals. Could you perhaps link me to one? Also, how can ...
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1answer
83 views

Partial Latin squares of even order

Can we show that $P$ is a partial latin square that is $n \times n$ where $n$ is even, where the upper quadrant $\frac{n}{2} \times \frac{n}{2}$ is filled and the rest is blank, then $P$ can be ...
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154 views

Overlaying Latin squares of order 4

Here are two latin squares overlayed upon each other to make one latin square, if you will. One "sub-latin" square is labeled with $1,2,3,4$ while the other is represented with $a,b,c,d$. There must ...
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Latin squares- completable-induction step [duplicate]

Possible Duplicate: Partial latin square with $\le n-1$ filled cells http://ajc.maths.uq.edu.au/pdf/22/ocr-ajc-v22-p247.pdf Could someone please further explain the inductive step here in ...
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145 views

Partial latin square with $\le n-1$ filled cells

How do we show that is $P$ if a $n\times n$ Latin square with $\le n-1$ filled cells, then $P$ can be completed to a proper Latin square? Here is the definition of a Latin square. (WHAT I HAVE DONE ...
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1answer
81 views

What is the length of a maximal deranged sequence of permutations

We were playing a home-made scribblish and were trying to figure out how to exchange papers. During each round, you'll trade k times and each time you need to give your current paper to someone who ...
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710 views

Formula for the number of latin squares of size $n$?

Is there a "easy to compute" formula for the number of Latin Squares or the number of reduced Latin squares of size $n$?
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1answer
113 views

Orthogonal Latin Squares

I'm not quite sure how to even start this problem. I'm really just looking for direction on how to begin. The $t$ mutually orthogonal Latin squares $A_1, A_2, ... , A_t$ of side $n$ have mutually ...
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3answers
466 views

Are there complete Graeco-Latin squares?

Are there two orthogonal complete Latin squares of any order greater than 1? If so, what is the smallest order for which they exist? (A Latin square of order $n$ is an $n\times n$ array of symbols ...
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66 views

Sorting numbers 1..6 to rows and columns [duplicate]

Possible Duplicate: Color an $n\times n$ square with $n$ colors How can I calculate the number of all possible permutations of a 6x6 grid having numbers 1 to 6 in every row and column, and ...
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3answers
173 views

Does this generalisation of Latin squares have a name?

I am interested in looking at $n\times n$ tableaux (or matrices) in which (WLOG) each integer in $\{ 1, 2, \ldots, n \}$ occurs exactly $n$ times. This is a generalisation of a Latin (or even ...
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798 views

Color an $n\times n$ square with $n$ colors

How many ways is there to color an $n\times n$ square grid with $n$ colors such that each column and each row contains exactly one $1\times 1$ square of each color? And how many ways if the same is ...
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2answers
155 views

square squares with diagonals also squares

The numbers reading across and down in these squares are square: $\begin{array}{ccc} 1 & 4 & 4\\ 4 &8&4\\ 4&4&1 \end{array}$ $\begin{array}{ccc} 5&2&9\\ ...
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Optimal Mixed Strategies for Latin Square Games

From Ferguson's Game Theory: A Latin square is an $n \times n$ array of $n$ different letters such that each letter occurs once and only once in each row and each column. Such games have simple ...
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Generate Random Latin Squares

I'm looking for algorithms to generate randomized instances of Latin squares. I found only one paper: M. T. Jacobson and P. Matthews, Generating uniformly distributed random Latin squares, J. ...