The fine structures of three pairwise distinct Latin squares

Let L k = ( a ij ( k ) ) v × v , k = 1 , 2 , 3 be Latin squares on symbol set S containing v elements. The fine structure of three Latin squares L 1 , L 2 , L 3 is defined to be ( t , s ) if s = v 2 - c 1 and t = c 2 , where c l is the number of cells ( i , j ) such that | { a ij ( 1 ) , a ij ( 2 ) , a ij ( 3 ) } | = l for l = 1 , 2 . Denote by Fin * ( v ) the set of all integer pairs ( t , s ) for which there exist three pairwise distinct Latin squares of order v on the same set having fine structure ( t , s ) . We determine the set Fin * ( v ) for any positive integer v ⩾ 8 .