▁natural ▁numbers ▁{ 1 ,..., v }. ▁Ch ose ▁and ▁fix , ▁in ▁order , ▁two ▁columns ▁of ▁A , ▁called ▁the ▁indexing ▁columns . ▁All ▁ordered ▁pairs ▁( i , ▁j ) ▁with ▁ 1 ▁ ≤ ▁i , ▁j ▁ ≤ ▁v ▁appear ▁exactly ▁once ▁in ▁the ▁rows ▁of ▁the ▁indexing ▁columns . ▁Take ▁any ▁other ▁column ▁of ▁A ▁and ▁create ▁a ▁square ▁array ▁whose ▁entry ▁in ▁position ▁( i , j ) ▁is ▁the ▁entry ▁of ▁A ▁in ▁this ▁column ▁in ▁the ▁row ▁that ▁contains ▁( i , ▁j ) ▁in ▁the ▁indexing ▁columns ▁of ▁A . ▁The ▁resulting ▁square ▁is ▁a ▁latin ▁square ▁of ▁order ▁v . ▁For ▁example , ▁consider ▁the ▁ 2 -( 3 , 4 , 1 ) ▁orthogonal ▁array : ▁ ▁By ▁choosing ▁columns ▁ 3 ▁and ▁ 4 ▁( in ▁that ▁order ) ▁as ▁the ▁indexing ▁columns , ▁the ▁first ▁column ▁produces ▁the ▁latin ▁square , ▁ ▁while ▁the ▁second ▁column ▁produces ▁the ▁latin ▁square , ▁ ▁The ▁latin ▁squares ▁produced ▁in ▁this ▁way ▁from ▁an ▁orthogonal ▁array ▁will ▁be ▁orthogonal ▁latin ▁squares , ▁so ▁the ▁k   −   2 ▁columns ▁other ▁than ▁the ▁indexing ▁columns ▁will ▁produce ▁a ▁set ▁of ▁k   −   2 ▁mut ually ▁orthogonal ▁latin ▁squares . ▁ ▁This ▁construction ▁is ▁completely ▁revers ible ▁and ▁so ▁strength ▁ 2 , ▁index ▁ 1 ▁orthogonal ▁arrays ▁can ▁be ▁constructed ▁from ▁sets ▁of ▁mut ually ▁orthogonal ▁latin ▁squares . ▁ ▁Latin ▁squares , ▁latin ▁cub es ▁and ▁latin ▁hyper c