For questions about latin squares, latin rectangles, their enumeration, their properties, generalisations and related combinatorial configurations such as MOLS (sets of Mutually Orthogonal Latin Squares).

A latin square of order $n$ is a configuration using $n$ symbols on a $n$ x $n$ square grid such as each line and each column contains each symbol exactly once.

A latin rectangle is a possibly incomplete latin square which could have fewer than $n$ lines or $n$ columns.

A latin cube is the same idea on $n$ x $n$ x $n$ grid

A special kind of latin square is the $9$x$9$ with additional constraints used in the "sudoku" game. In this case, the square is further divided in nine $3$x$3$ subsquares required to contain once all $9$ symbols.

Latin squares are the combinatorial basis of finite quasigroups and their special cases (such as loops = quasigroups with a neutral element). The symbols represent the elements of these algebraic structures and the content of the square give the result of the internal binary operation between these symbols. As such they have several relations to finite group theory (the Cayley table of a finite group is a latin square). They can also be seen as two-dimensional permutations. Each row and each column of a latin square is a permutation of the list of symbols, and there is no fixed point in the permutations transforming any column into any other or in the permutations transforming any row into any other.

Enumeration of latin squares and rectangles and related questions is an active research topic. Wikipedia - problems in latin squares

Mutually Orthogonal Latin Squares

Two latin squares of the same order on two different sets of symbols are mutually orthogonal if, when superposed, each combination of two symbols in each cell is unique.

An important theorem of Bose and Shrikhande (1959), disproving a conjecture of Euler is that:

There exists mutually orthogonal sets of latin squares (MOLS) of order $n$ and number of squares at least $2$ for every $n$ greater than $2$ and different from $6$.

MOLS are used in design of experiments (see also factorial designs), are related to projective planes and several other types of discrete geometry.

See also: Euler 36 officers problem, Graeco-Latin squares, combinatorial block designs, tournaments, projective planes, derangements.