A Latin square is a n X n matrix with n different elements, placed so that each row and column contain each element exactly once (Sudoku's are a special case of these)
Two latin squares are orthogonal when you pair up corresponding elements and get exactly one of each pair.
Let say we have a 4X4 matrix:
a b c d
b a d c
c d a b
d c b a
and another:
1 2 3 4
3 4 1 2
4 3 2 1
2 1 4 3
and combined matrix would be:
(a, 1) (b, 2) (c, 3) (d, 4)
(b, 3) (a, 4) (d, 1) (c, 2)
(c, 4) (d, 3) (a, 2) (b, 1)
(d, 2) (c, 1) (b, 4) (a, 3)
From here we see that the two matrices (or latin squares) are orthogonal, because each pair ((a, 1) f'rinstance) exists exactly once. It's also easy to see with more primitive visual aids that no latin square is orthogonal with itself (because each pair would be found exactly n times).
I hope this is helpful.
Does anyone know it orthogonality is transitive? I forget. |
