formula for the number of Latin squares is where is the set of all {0, 1}-matrices, is the number of zero entries in matrix , and is the permanent of matrix . The table below contains all known exact values. It can be seen that the numbers grow exceedingly quickly. For each , the number of Latin squares altogether is times the number of reduced Latin squares . For each , each isotopy class contains up to Latin squares (the exact number varies), while each main class contains either 1, 2, 3 or 6 isotopy classes. The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflection, and/or permutation of the symbols) for = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively . Examples We give one example of a Latin square from each main class up to order five. They present, respectively, the multiplication tables of the following groups: {0} – the trivial 1-element group – the binary group – cyclic group of order 3 – the Klein four-group – cyclic group of order 4 – cyclic group of order 5 the last one is an example of a quasigroup, or rather a loop, which is not associative. Transversals and rainbow matchings A transversal in a Latin square is a choice of n cells, where each row contains one cell, each column contains one cell, and there is one cell containing each symbol. One can consider a Latin square as a complete bipartite graph in which the rows are vertices of one part, the columns are vertices of the other part, each cell is an edge (between its row and its column), and the symbols are colors. The rules of the Latin squares imply that this is a proper edge coloring. With this definition, a Latin transversal is a matching in which each edge has a different color; such a matching is called a rainbow matching. Therefore, many results on Latin squares/rectangles are contained in papers with the term "rainbow matching" in their title, and vice versa. Some Latin squares have no transversal. For example, when n is even, an n-by-n Latin square in which the value of cell i,j is (i+j) mod n has no transversal. Here are two examples:In 1967, H. J. Ryser conjectured that, when n is odd, every n-by-n Latin square has