4 0 0 ▁possibilities . ▁They ▁also ▁provide ▁good ▁examples ▁for ▁back track ing ▁parad ig m ▁of ▁algorithm ▁design . ▁ ▁Other ▁information ▁When ▁generalized ▁to ▁arbitrary ▁bases , ▁the ▁problem ▁of ▁determ ining ▁if ▁a ▁crypt arith m ▁has ▁a ▁solution ▁is ▁N P - complete . ▁( The ▁general ization ▁is ▁necessary ▁for ▁the ▁hard ness ▁result ▁because ▁in ▁base ▁ 1 0 , ▁there ▁are ▁only ▁ 1 0 ! ▁possible ▁assign ments ▁of ▁digits ▁to ▁letters , ▁and ▁these ▁can ▁be ▁checked ▁against ▁the ▁puzz le ▁in ▁linear ▁time .) ▁ ▁Al ph am et ics ▁can ▁be ▁combined ▁with ▁other ▁number ▁puzz les ▁such ▁as ▁Sud oku ▁and ▁K ak uro ▁to ▁create ▁crypt ic ▁Sud oku ▁and ▁K ak uro . ▁ ▁Long est ▁al ph am et ics ▁ ▁Anton ▁Pav lis ▁constructed ▁an ▁al ph am etic ▁in ▁ 1 9 8 3 ▁with ▁ 4 1 ▁add ends : ▁ ▁SO + MAN Y + MO RE + M EN + SE EM + TO + SA Y + TH AT + ▁THE Y + MA Y + SO ON + TR Y + TO + ST A Y + AT + HOME + ▁SO + AS + TO + SEE + OR + HE AR + THE + SA ME + ONE + ▁M AN + TR Y + TO + ME ET + THE + TE AM + ON + THE + ▁MO ON + AS + HE + H AS