ub es ▁ ▁Orth ogonal ▁arrays ▁provide ▁a ▁uniform ▁way ▁to ▁describe ▁these ▁diverse ▁objects ▁which ▁are ▁of ▁interest ▁in ▁the ▁statistical ▁design ▁of ▁experiments . ▁ ▁Latin ▁squares ▁ ▁As ▁mentioned ▁in ▁the ▁previous ▁section ▁a ▁latin ▁square ▁of ▁order ▁n ▁can ▁be ▁thought ▁of ▁as ▁a ▁ 2 -( n , ▁ 3 , ▁ 1 ) ▁orthogonal ▁array . ▁Actually , ▁the ▁orthogonal ▁array ▁can ▁lead ▁to ▁six ▁latin ▁squares ▁since ▁any ▁ordered ▁pair ▁of ▁distinct ▁columns ▁can ▁be ▁used ▁as ▁the ▁indexing ▁columns . ▁However , ▁these ▁are ▁all ▁is ot op ic ▁and ▁are ▁considered ▁equivalent . ▁For ▁con cre ten ess ▁we ▁shall ▁always ▁assume ▁that ▁the ▁first ▁two ▁columns ▁in ▁their ▁natural ▁order ▁are ▁used ▁as ▁the ▁indexing ▁columns . ▁ ▁Latin ▁cub es ▁ ▁In ▁the ▁statistics ▁literature , ▁a ▁latin ▁cube ▁is ▁an ▁n ▁× ▁n ▁× ▁n ▁three - dimensional ▁matrix ▁consisting ▁of ▁n ▁layers , ▁each ▁having ▁n ▁rows ▁and ▁n ▁columns ▁such ▁that ▁the ▁n ▁distinct ▁elements ▁which ▁appear ▁are ▁repeated ▁n 2 ▁times ▁and ▁arranged ▁so ▁that ▁in ▁each ▁layer ▁parallel ▁to ▁each ▁of ▁the ▁three ▁pairs ▁of ▁opposite ▁faces ▁of ▁the ▁cube ▁all ▁the ▁n ▁distinct ▁elements ▁appear ▁and ▁each ▁is ▁repeated ▁exactly ▁n ▁times ▁in ▁that ▁layer . ▁ ▁Note ▁that ▁with ▁this ▁definition ▁a ▁layer ▁of ▁a ▁latin ▁cube ▁need ▁not ▁be ▁a ▁latin ▁square . ▁In ▁fact , ▁no ▁row , ▁column ▁or ▁file ▁( the ▁cells ▁of ▁a ▁particular ▁position ▁in ▁the ▁different ▁layers )