▁paths ▁are ▁Hamiltonian , ▁with ▁a ▁size ▁of ▁ 2 , 7 0 7 , 0 7 5 . ▁Z DD s ▁in ▁this ▁case , ▁are ▁efficient ▁for ▁simple ▁paths ▁and ▁Hamiltonian ▁paths . ▁ ▁the ▁E ight - que ens ▁problem ▁ <0x09> ▁Define ▁ 6 4 ▁input ▁variables ▁to ▁represent ▁the ▁squares ▁on ▁a ▁ch ess ▁board . ▁Each ▁variable ▁denotes ▁the ▁presence ▁or ▁absence ▁of ▁a ▁queen ▁on ▁that ▁square . ▁Consider ▁that , ▁▁ <0x09> In ▁a ▁particular ▁column , ▁only ▁one ▁variable ▁is ▁“ 1 ”. ▁ <0x09> In ▁a ▁particular ▁row , ▁only ▁one ▁variable ▁is ▁“ 1 ”. ▁ <0x09> On ▁a ▁particular ▁diagonal ▁line , ▁one ▁or ▁no ▁variable ▁is ▁“ 1 ”. ▁Although ▁one ▁can ▁solve ▁this ▁problem ▁by ▁construct ing ▁O B DD s , ▁it ▁is ▁more ▁efficient ▁to ▁use ▁Z DD s . ▁Con struct ing ▁a ▁Z DD ▁for ▁the ▁ 8 - Que ens ▁problem ▁requires ▁ 8 ▁steps ▁from ▁S 1 ▁to ▁S 8 . ▁Each ▁step ▁can ▁be ▁defined ▁as ▁follows : ▁ <0x09> S 1 : ▁Re pres ents ▁all ▁choices ▁of ▁putting ▁a ▁queen ▁at ▁the ▁first ▁row . ▁S 2 : ▁Re pres ents ▁all ▁choices ▁of ▁putting ▁a ▁queen ▁at ▁the ▁second ▁row ▁so ▁as ▁not ▁to ▁viol ate ▁the ▁first ▁queen . ▁ ▁S 3 : ▁Re pres ents ▁all ▁choices ▁of ▁putting ▁a ▁queen ▁at ▁the ▁third ▁row ▁so ▁that ▁it ▁does ▁not ▁viol ate ▁the ▁previous ▁que ens .