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I just read The Nine Billion Names of God by Arthur C. Clarke.

In that story, the monks estimated (or confirmed) the number of 9,000,000,000(nine billion) for all the possible names of God with nine characters in max.

That contains, I think, (using English alphabets)

0          a
           b
           c
           ...
           aa
           bb
           ...
           ccc
           ...
9000000000 zzzzzzzzz (whatever the last alphabet)

Now how can I calculate (or estimate) the total number of the monks' alphabets whose all nine (or less) combinations counts 9,000,000,000?

Jin Kwon
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    Assuming the English alphabet of $26$ letters (case in-sensitive), the number of possible strings of length $1$ to $9$ is equal to $\sum\limits_{i=1}^9 26^i\approx 5,646,000,000,000$. If you are asking how many letters are in the monks' alphabet to have arrived at a number of approximately $9$ billion, with $12$ letters that would be approximately $5.6$ billion while with $13$ letters that would be approximately $11.5$ billion. – JMoravitz Jun 14 '19 at 12:10
  • Oh my dear @JMoravitz, please make an answer so that I can accept it. – Jin Kwon Jun 14 '19 at 12:14
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    As it is phrased, I don't think that this is really a question about mathematics---you have not sufficiently nailed down the question (are you asking "If an alphabet can spell out 9 billion words, how many letters must be in that alphabet?" or are you asking "How many different subsets of the English alphabet can be used to spell out 9 billion words?"). Also, as I recall, Clarke suggests that not every possible combination of letters is a name of god, so only a finite subset of the possible words count---should we account for this? How? – Xander Henderson Jun 14 '19 at 12:17
  • @XanderHenderson I'm sorry I'm not good in English neither in Math. I wanted to know, in a finite set of alphabets, all number of unique words, which each word's length is equals or less than 9, (approximately) is 9 billion, how many characters required to be defined in that alphabet. – Jin Kwon Jun 14 '19 at 12:23
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    My memory is that (some / all?) letters could occur three times successively but no more. I don't recall whether there were other constraints or whether it was claimed that there were no others. My best guess is that the exact rules were left vague. – badjohn Jun 14 '19 at 12:28
  • Oh, God. I realized I forgot the constraint. I don't know the exact literal condition of the constraint cuz I read it in Korean-translated. Can anybody please quote the constraint in original form? – Jin Kwon Jun 14 '19 at 12:48
  • I found it. A rather more interesting problem is that of devising suitable circuits to eliminate ridiculous combinations. For example, no letter must occur more than three times in succession.” I should've known that and mentioned that. Sorry. – Jin Kwon Jun 14 '19 at 12:53

1 Answers1

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The number of strings of length exactly $k$ that can be formed from an alphabet of $n$ characters is $n^k$, seen by direct application of the rule of product.

The number then of non-empty strings of length at most $k$ that can be formed from an alphabet of $n$ characters is $\sum\limits_{i=1}^k n^i$

Plugging in some numbers, we have the number of possible non-empty (case-insensitive) strings using the $26$ letters from the English alphabet of length at most $9$ will be:

$$\sum\limits_{i=1}^9 26^i \approx 5,646,000,000,000$$

As for how many letters are in the monks' alphabet to have arrived at a value of $9$ billion, you have $12$ letters would give you approximately $5.6$ billion while $13$ letters would give you approximately $11.5$ billion possible non-empty strings of length at most $9$.

Of course, this entirely ignores certain constraints such as the number of consecutive consonants or placements of vowels and the like, so many of these strings would be unpronouncable.

JMoravitz
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  • So, besides those linguistic problems, there should be at least 13 characters in that alphabet, right? Thanks. – Jin Kwon Jun 14 '19 at 12:27
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    As I said in a comment to question, there was a rule that no letter occurred more than three consecutively. I don't recall whether there were more rules. – badjohn Jun 14 '19 at 12:30