▁need ▁be ▁a ▁permut ation ▁of ▁the ▁n ▁symbols . ▁ ▁A ▁latin ▁cube ▁of ▁order ▁n ▁is ▁equivalent ▁to ▁a ▁ 2 -( n , ▁ 4 , ▁n ) ▁orthogonal ▁array . ▁ ▁Two ▁latin ▁cub es ▁of ▁order ▁n ▁are ▁orthogonal ▁if , ▁among ▁the ▁n 3 ▁pairs ▁of ▁elements ▁chosen ▁from ▁corresponding ▁cells ▁of ▁the ▁two ▁cub es , ▁each ▁distinct ▁ordered ▁pair ▁of ▁the ▁elements ▁occurs ▁exactly ▁n ▁times . ▁ ▁A ▁set ▁of ▁k   −   3 ▁mut ually ▁orthogonal ▁latin ▁cub es ▁of ▁order ▁n ▁is ▁equivalent ▁to ▁a ▁ 2 -( n , ▁k , ▁n ) ▁orthogonal ▁array . ▁ ▁An ▁example ▁of ▁a ▁pair ▁of ▁mut ually ▁orthogonal ▁latin ▁cub es ▁of ▁order ▁three ▁was ▁given ▁as ▁the ▁ 2 -( 3 , 5 , 3 ) ▁orthogonal ▁array ▁in ▁the ▁Ex amples ▁section ▁above . ▁ ▁Un like ▁the ▁case ▁with ▁latin ▁squares , ▁in ▁which ▁there ▁are ▁no ▁constraints , ▁the ▁indexing ▁columns ▁of ▁the ▁orthogonal ▁array ▁representation ▁of ▁a ▁latin ▁cube ▁must ▁be ▁selected ▁so ▁as ▁to ▁form ▁a ▁ 3 -( n , 3 , 1 ) ▁orthogonal ▁array . ▁ ▁Latin ▁hyper c ub es ▁ ▁An ▁m - dimensional ▁latin ▁hyper c ube ▁of ▁order ▁n ▁of ▁the ▁r th ▁class ▁is ▁an ▁n ▁× ▁n ▁× ▁... ▁× n ▁m - dimensional ▁matrix ▁having ▁nr ▁distinct ▁elements , ▁each ▁repeated ▁n m   −   r ▁times , ▁and ▁such ▁that ▁each ▁element ▁occurs ▁exactly