Sunday, 6 April 2014

no computers - If 6 was 9, or 100 was 64, or M was N


         Now if a 6,
         Turned out to be 9,

         I don’ t mind,
         I don’ t mind   – Jimi Hendrix in If 6 was 9


Cool. Suppose a 100 turned out to be 64.


         I don’ t care,
         I don’ t care   – Jimi Hendrix in If 6 was 9


But, but Jimi, what if your computer program is bugged out and can’t tell if its bad self is coming (from octal) or going (to hex)?


          1008    =  6410
          10010  =  6416
          10016  =  6442


         Let it be,

         It ain’ t me   – Jimi Hendrix in If 6 was 9


You know, 1008 stands for the digits 100 in base 8, equaling the familiar 64 (in base 10), which itself shows as 6410.


         Dig    – Jimi Hendrix in If 6 was 9


So, that used 6 different digits — 0, 1, 2, 4, 6, 8 — a total of 26 times and bubbles down to...


         K  W  =  L X
         K X   =  L Y
         K Y   =  L Z
         ...where   K = 100,   L = 64,   W = 8,   X = 10,   Y = 16   and   Z = 42.


         I’ m gonna wave my freak flag high! – Jimi Hendrix in If 6 was 9




       Hold tight, hitch-hiker, check out where simpler ingredients can take you.




Just 4 different digits from 0 through 9, taken fewer than 43 times total, can really wig you out.


         M P   =  N Q     ( M > N   and   P < Q < R < S < T )
         M Q  =  N R
         M R  =  N S
         M S   =  N T


     Wild.   What M, N, P, Q, R, S and T could do that with the fewest digits total?


         Even a 43-digits-total solution or two would be worth posting.




         Wave on,
         Wave on . . .
         . . .’ Cause everybody knows what I’ m talking about
          – Jimi Hendrix in If 6 was 9 (vocal track) dailymotion



Answer



Well, I have a working set, it seems:



1011 (base 1) = 11 (base 2)
1011 (base 2) = 11 (base 10)

1011 (base 10) = 11 (base 1010)
1011 (base 1010) = 11 (base 1030302010)



using 4 digits (0,1,2 and 3) and the total number of times those numbers used is - 49 times (well, it was above the 43 limit...) and


assuming fractional bases are valid, here is another one:



22 (base 2.25) = 11 (base 5.5)
22 (base 5.5) = 11 (base 12)
22 (base 12) = 11 (base 25)
22 (base 25) = 11 (base 51 )




using 3 digits (1, 2 and 5) and the total number of times those numbers used is 33.


Here is another one:



111 (base 2) = 21 (base 3)
111 (base 3) = 21 (base 6)
111 (base 6) = 21 (base 21)
111 (base 21) = 21 (base 231)



using 4 digits (1, 2, 3 and 6) and the total number of times those numbers used is 32. Hope this satisfies @humn's requirements in all manners.



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