Articles of latin square

How many structurally distinct Latin squares are there of order 7?

What is the 7th term of 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?