▁only ▁two ▁pairs ▁of ▁available ▁numbers ▁that ▁sum ▁to ▁at ▁least ▁ 1 2 ▁are ▁( 5 , 7 ) ▁and ▁( 6 , 7 ) ▁so ▁either ▁E ▁= ▁ 7 ▁or ▁D ▁= ▁ 7 . ▁Since ▁N ▁= ▁E ▁+ ▁ 1 , ▁E ▁can ' t ▁be ▁ 7 ▁because ▁then ▁N ▁= ▁ 8 ▁= ▁R ▁so ▁D ▁= ▁ 7 . ▁E ▁can ' t ▁be ▁ 6 ▁because ▁then ▁N ▁= ▁ 7 ▁= ▁D ▁so ▁E ▁= ▁ 5 ▁and ▁N ▁= ▁ 6 . ▁D ▁+ ▁E ▁= ▁ 1 2 ▁so ▁Y ▁= ▁ 2 . ▁ ▁The ▁use ▁of ▁mod ular ▁arithmetic ▁often ▁helps . ▁ ▁For ▁example , ▁use ▁of ▁mod - 1 0 ▁arithmetic ▁allows ▁the ▁columns ▁of ▁an ▁addition ▁problem ▁to ▁be ▁treated ▁as ▁simultane ous ▁equations , ▁while ▁the ▁use ▁of ▁mod - 2 ▁arithmetic ▁allows ▁infer ences ▁based ▁on ▁the ▁par ity ▁of ▁the ▁variables . ▁ ▁In ▁computer ▁science , ▁crypt arith ms ▁provide ▁good ▁examples ▁to ▁illustrate ▁the ▁br ute ▁force ▁method , ▁and ▁algorithms ▁that ▁ ▁generate ▁all ▁permut ations ▁of ▁m ▁choices ▁from ▁n ▁possibilities . ▁For ▁example , ▁the ▁D uden ey ▁puzz le ▁above ▁can ▁be ▁solved ▁by ▁testing ▁all ▁assign ments ▁of ▁eight ▁values ▁among ▁the ▁digits ▁ 0 ▁to ▁ 9 ▁to ▁the ▁eight ▁letters ▁S , E , N , D , M , O , R , Y , ▁giving ▁ 1 , 8 1 4 ,