Articles of latin square

Can a sudoku with valid columns and rows be proved valid without evaluating every 3×3 inside it?

I’m trying to solve a computer science challenge and have readily been able to validate whether or not the outside dimensions of a sudoku puzzle are valid. However, it doesn’t check the validity of the inside squares and will “validate” the following incorrect sudoku puzzle because the 3×3 squares are not unique 1-9: [1, 2, […]

Orthogonal Latin Square 6*6

I need to make remarks about Tarry’s Proof for the nonexistence of 6×6 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 minor comments about it that don’t explain what he did. Does anyone know where I […]

Transversals of Latin Squares

According to this thesis, page $28$, the following Latin Square has $3$ $0$-s transversals: $$\begin{bmatrix}1 & 2 & 3 & 4 & 5\\ 2 & 4 & 1 & 5 & 3\\ 3 & 5 & 4 & 2 & 1\\ 4 & 1 & 5 & 3 & 2\\5 & 3 & 2 & […]

Algorithms for mutually orthogonal latin squares – a correct one?

I am very interested in using mutually orthogonal latin squares (MOLS) to reduce the number of test cases but I struggle to find a way how to implement the algorithm. In an article published in a foreign language, I did find a formula including the source code. This is said to be a formula to […]

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 of rotation, reflection, and/or permutation of the symbols.

Find combinations of N sets having no more than one item in common.

Problem definition; There are N input sets, of sizes $S_1, S_2, … ,S_N$. eg; 4 sets – (1a,2a,3a,4a,5a), (1b,2b,3b,4b), (1c,2c,3c), (1d,2d,3d) A combination is made by selecting one item from each input set. eg; [1a, 1b, 1c, 1d] Rule: No chosen combination may have more than one item in common with any other. eg if […]

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 the exchanges) ?

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 think of it as: Theorem: For, $k \in \{0,1,\ldots,n-1\}$, any $k \times n$ Latin rectangle extends to a $(k+1) \times n$ Latin rectangle. The only […]

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. Combinatorial Design 4 (1996), 405-437 Is there any other method known which generates truly random instances not just isomorphic instances of other instances?

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 required of the two diagonals?