A Latin square L is called self-orthogonal if L and its transpose Ltr form an orthogonal pair.

Question:

A Latin square L is called self-orthogonal if L and its transpose Ltr form an orthogonal pair.
a) Show that there is no 3 × 3 self-orthogonal Latin square.
b) Give an example of a 4 × 4 Latin square that is self- orthogonal.
c) If L = (aij) is an n × n self-orthogonal Latin square, prove that the elements aii, for 1 < i < n, must all be distinct.