Thursday, November 3, 2016

lateral thinking - Finger the digit


Officials acquired two anonymized self texts by the G Gang member known as i, and hope these will help you identify the boss, B.




$\kern166mu$ G=i N t E r M s O f G

$\kern199mu$ . G=i   /   B . i



Investigators have thus far determined that each gangster (iN, t, E, r, M, s, O, f and B) must secretly be one of the foremost citizens of Whollywood (01, 2, 3, 4, 5, 6, 7, 8 or 9) and that foul mathplay is surely afoot.



Just which digit do you finger as B?


( No leading zeros in Whollywood.   G is not an 11th digit.   =, . and / are not in disguise. )



Answer



The boss B is



8



Since G is defined in terms of itself, it stands to reason that



G is an infinitely repeating sequence of digits




Since B is missing from G, we can figure that .G is



$.G = 1 / 8.1 =\ .\overline{123456790}$

(Could be figured out via fractions of repeated 9s, as in: 1/9=.111..., 12/99=.121212..., 475/999=.475475475..., iNtErMsOf/999999999 = .iNtErMsOf iNtErMsOf iNtErMsOf .... The divisors of 999999999 are easy because it is so divisible by 3: 999999999=3×3×3×3×12345679.)



and G is then



$\overline{123456790}$



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