For questions on or pertaining to Latin squares.

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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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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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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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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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38 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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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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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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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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131 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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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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283 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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85 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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337 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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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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133 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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161 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
105 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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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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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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143 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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66 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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129 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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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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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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369 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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211 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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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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51 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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114 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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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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137 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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186 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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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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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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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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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 ...