7
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The linguistic hardness ($LH$) of a natural number is the ratio of the amount of letters in the writing of this number in English to the amount of its digits.

For example, $LH(1234) = 7.75$, as:

$$\frac{\mbox{Number of letters}}{\mbox{Number of digits}} = \frac{\mbox{N(one thousand two hundred thirty four)}}{N(1234)}= 31/4 = 7.75$$

What the is $N: 0<N<10^9$ with biggest $LH(N)$?

P.S. This puzzle appears to be much more interesting in Russian, but this is not Russian site to ask it here.

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  • $\begingroup$ Is the solution unique? $\endgroup$ – 355durch113 Aug 1 '14 at 0:07
  • $\begingroup$ @Carlster, I don't know the solution. $\endgroup$ – klm123 Aug 1 '14 at 4:26
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    $\begingroup$ There could be regional differences to this calculation since, for example, British conventions call for an "and" between 10^2 and 10^1, while North American don't. c.f. $\endgroup$ – user2096 Aug 14 '14 at 13:34
  • $\begingroup$ It would be more interesting if we were comparing the letter count with the logarithm (rather than digit count) of the number. $\endgroup$ – AJMansfield Jan 9 '15 at 2:51
14
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For any given digit range the numbers with the longest spelling have the highest LH.
7-only-combinations are always among those numbers. Here's a table with their corresponding LH:

 number from here | number segment | letters from here | LH
------------------+----------------+-------------------+-------
        777777777 | seven hundred  | 87                | 9 2/3 = 9 14/21
         77777777 | seventy        | 75                | 9 3/8
          7777777 | seven million  | 68                | 9 5/7 = 9 15/21
           777777 | seven hundred  | 56                | 9 1/3
            77777 | seventy        | 44                | 8 4/5
             7777 | seven thousand | 37                | 9 1/4
              777 | seven hundred  | 24                | 8
               77 | seventy        | 12                | 6
                7 | seven          | 5                 | 5

Looks like 7777777 beat my previous suggestion. It worries me that LH(8878878) is the same (alongside others like 3878373).

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    $\begingroup$ This is a cheap way to increase the letter count, but you could write "million" as "thousand thousands". I wouldn't actually do that, but you could do it. $\endgroup$ – Duncan Jul 31 '14 at 23:43
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    $\begingroup$ This puzzle appears to be much more interesting in Russian, unfortunately this is not Russian site to ask it here. $\endgroup$ – klm123 Jan 3 '15 at 9:33
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    $\begingroup$ @klm123 nothing wrong with posting an additional Russian answer plus explaining why it is more 'interesting'. I'd love to see that... $\endgroup$ – BmyGuest Jan 5 '15 at 9:27
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    $\begingroup$ A smaller number with just as many letters would be 373373373, because "three" is just as long as "seven". The advantage to seven is only in "seventy" versus "thirty". $\endgroup$ – AJMansfield Jan 9 '15 at 2:49

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