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a) is easy, b) is not (Posted on 2011-02-18) Difficulty: 3 of 5
1, 2, 4, 8, 1, 3, 6, 1 ... is a non-cyclic series where a(n) represents the leading digit of 2^n.

a) How many ones are there within the first 3000 members?
b) Same question for a digit d, other than digit one.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (2 votes)

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Not an approximation? | Comment 7 of 8 |
Well, I guess my approximation to part (a) wasn't really an approximation.  As Jer points out, the number of ones is necessarily the same as the number of digits in 2^3000, and this is exactly equal to 3000*log 2, rounded up to the next integer.  Benford's law is still an approximation for the other values of d, although it is a pretty good one.

Benford's law:
                predicted  actual per xdog
1 30.1%               904
2 17.6%   528       529
3 12.5%   375       374
4 9.7%     291       291
5 7.9%     237       238
6 6.7%     201       201
7 5.8%     174       173 
8 5.1%     153       155
9 4.6%     138       136

  Posted by Steve Herman on 2011-02-19 15:47:20
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