# 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? Aug 1, 2014 at 0:07
• @Carlster, I don't know the solution. Aug 1, 2014 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, 2014 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. Jan 9, 2015 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. Jul 31, 2014 at 23:43
• This puzzle appears to be much more interesting in Russian, unfortunately this is not Russian site to ask it here. Jan 3, 2015 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... Jan 5, 2015 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". Jan 9, 2015 at 2:49