# Tagged Questions

For questions on or pertaining to Latin squares.

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### How to generate centrally symmetric Latin squares?

I'd like to generate centrally symmetric latin squares for arbitrary sizes. For example for size 6 it could be ...
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### The Hardest Sudoku Puzzle

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers ...
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### Counting the number of Latin squares

Counting the number of latin squares is a difficult problem. I understand that the common used formula is $n!(n-1)!$ (the number or reduced latin squares). As seen here and in many other pages you can ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...
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 \$n=a\...