# What is the “linguistically hardest” number less than $10^9$?

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.

• Is the solution unique? – 355durch113 Aug 1 '14 at 0:07
• @Carlster, I don't know the solution. – klm123 Aug 1 '14 at 4:26
• 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. – user2096 Aug 14 '14 at 13:34
• It would be more interesting if we were comparing the letter count with the logarithm (rather than digit count) of the number. – AJMansfield Jan 9 '15 at 2:51

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).

• 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. – Duncan Jul 31 '14 at 23:43
• This puzzle appears to be much more interesting in Russian, unfortunately this is not Russian site to ask it here. – klm123 Jan 3 '15 at 9:33
• @klm123 nothing wrong with posting an additional Russian answer plus explaining why it is more 'interesting'. I'd love to see that... – BmyGuest Jan 5 '15 at 9:27
• 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". – AJMansfield Jan 9 '15 at 2:49