A latin square or rectangle is a square in which the rows and columns do
not have a repeat number in them (similar to
distributive magic squares, only with less different numbers). A sudoku is also a form of Latin
square, as the rows and columns never have a double number in them. For
Latin squares only the numbers 1,2,...n are allowed to be used, with n
being the order of the square. Therefore, for order 2 there are 2 possible
Latin squares and for order 3 there are 12. Orthogonal Latin squares occur
when two Latin squares are overlaid and they form distinct pairs in each
position of the square and still keep their Latin square feature of not
having a recurring number in each row and column. This is also called a
Euler square or Greaco-Latin, Greaco-Roman, Latin-Greaco or mutually
orthogonal Latin squares. Euler squares exist for n=3,4 and for every
other order n expect n=2 and n=6. This was discovered in 1959, as before
they believed that every n=3k and n=4k+2 (k=1,2,...n) order Euler magic
square was impossible.
