Complete the world's longest self-documenting sentence

Question

Complete the following by filling in the blanks with written numbers to create the world's longest self-documenting sentence.

This sentence uses or mentions
    the word "a" ___ time(s),
    the word "aah" ___ time(s),
    the word "aahed" ___ time(s),
        ...
    and the word "zyzzyvas" ___ time(s).

A few notes:

• The "..." omits a lot. There should be one line of the form "the word 'blah' __ time(s)" for each word in this word list that contains 109582 words.
• Numbers should be written out. For example, 109582 should be written "one hundred nine thousand five hundred eighty-two". The "eighty-two" counts as two different words.
• The uses or mentions is important: a word should be counted whether it is in quotation marks or not.
• For the final sentence, each "time(s)" should be either "time" or "times".
• I wouldn't recommend writing your answer completely out. Just give us the important bits.
• I'm not actually sure if it is the world's longest, but it's a pretty good stab at it.

Let me know if I can clarify anything else.

Answer 1

Solution

This sentence uses or mentions
    the word "a" one time,
        ...
    the word "and" two times,
        ...
    the word "eighty" three times,
        ...
    the word "five" six times,
        ...
    the word "four" three times,
        ...
    the word "hundred" nine times,
        ...
    the word "mentions" two times,
        ...
    the word "nine" seven times,
        ...
    the word "one" one hundred nine thousand five hundred sixty four times,
        ...
    the word "or" two times,
        ...
    the word "sentence" two times,
        ...
    the word "seven" three times,
        ...
    the word "six" two times,
        ...
    the word "sixty" three times,
        ...
    the word "the" one hundred nine thousand five hundred eighty three times,
        ...
    the word "this" two times,
        ...
    the word "thousand" five times,
        ...
    the word "three" seven times,
        ...
    the word "time" one hundred nine thousand five hundred sixty times,
    the word "times" twenty four times,
        ...
    the word "twenty" two times,
        ...
    the word "two" nine times,
        ...
    the word "uses" two times,
        ...
    the word "word" one hundred nine thousand five hundred eighty three times,
        ...
    and the word "zyzzyvas" one time.

(where each ... denotes a whole string of lines of the form "the word "[...]" one time,").

---

Working

The words "this", "sentence", "uses", "or", "mentions", and "and" are each going to be used exactly two times (once for the places we can see them in the OP, once for their appearance in the list of words in quotes).

The words "the" and "word" are each going to be used exactly one hundred nine thousand five hundred eighty three times (once for every word in the list, and once more for their own appearances in the list).

The only other words that will be used more than once are "time", "times", and various number-related words ("one", "hundred", "nine", etc.) Let $n$ be the number of words that will be used more than once; $n$ is going to be less than thirty.

The word "times" is going to be used $n+1$ times (once for each word used more than once, once for its own appearance in the list). The word "time" is going to be used $109582-n+1$ times (once for each word used exactly once, once for its own appearance in the list).

The only words which will be used more than $n+1$ times are "the", "word", "one", and "time", which each appear at least $109582-n+1$ times. So the word "hundred" is going to be used nine times, the word "thousand" five times, and the word "eighty" three times.

The bolded numbers above tell us that "one", "nine", "five", "two", and "three" will each be used more than once. So already $n$ is at least eighteen. Assuming $n$ is at least nineteen and less than twenty-nine (which I'm pretty sure it will be), the word "twenty" is used two times (once for the number of times "times" is used and once for its own appearance in the list).

So far we have the following words appearing more than once:

• "this" ($2$ times)
• "sentence" ($2$ times)
• "uses" ($2$ times)
• "or" ($2$ times)
• "mentions" ($2$ times)
• "and" ($2$ times)
• "the" ($109583$ times)
• "word" ($109583$ times)
• "times" ($n+1$ times)
• "time" ($109582-n+1$ times)
• "one" ($\geq109582-n+1+4$ times)
• "hundred" ($9$ times)
• "thousand" ($5$ times)
• "eighty" ($3$ times)
• "nine" ($\geq6$ times)
• "five" ($\geq6$ times)
• "two" ($\geq8$ times)
• "three" ($\geq4$ times)
• "twenty" ($2$ times)

---

Let's now assume $n\leq23$, so that "time" is used between one hundred nine thousand five hundred sixty and one hundred nine thousand five hundred sixty four times. Now the word "sixty" is going to be used either two or three times (according to whether or not "one" is used as much as 109570 times), so $n\geq20$.

If $n=20$, then "times" is used 21 times, "time" is used 109563 times, and so "one" is used 109568 times, which means "eight" is used more than once and $n\geq21$, contradiction.

If $n=21$, then "times" is used 22 times, "time" is used 109562 times, and so "one" is used 109566 times, which means "six" is used more than once and is the 21st and last such word. But "two" is used at least 10 times (and fewer than 20), which means one of the words "ten", "eleven", ... "nineteen" must be used more than once, contradiction.

---

If $n=22$, then "times" is used 23 times, "time" is used 109561 times, and so "one" is used 109566 times, which means "six" is used more than once. Now "sixty" is used exactly 3 times and "two" is used 8, 9, or 10 times.

If "two" is used 8 times, then both "six" and "eight" must be used more than 2 times and we have:

• "this" ($2$ times)
• "sentence" ($2$ times)
• "uses" ($2$ times)
• "or" ($2$ times)
• "mentions" ($2$ times)
• "and" ($2$ times)
• "the" ($109583$ times)
• "word" ($109583$ times)
• "times" ($23$ times)
• "time" ($109561$ times)
• "one" ($109566$ times)
• "hundred" ($9$ times)
• "thousand" ($5$ times)
• "eighty" ($3$ times)
• "nine" ($\geq6$ times)
• "five" ($\geq6$ times)
• "two" ($8$ times)
• "three" ($\geq6$ times)
• "twenty" ($2$ times)
• "sixty" ($3$ times)
• "six" ($\geq3$ times)
• "eight" ($\geq3$ times)

Since we have 22 words in this list, we can have no more. So each of "nine", "five", and "three" must be used 6 or 8 times while each of "six" and "eight" must be used exactly 3 times, contradiction.

If "two" is used 10 times, then each of "six" and "ten" must be used exactly 2 times. So "three" is used six times, which means "six" is used at least 3 times, contradiction.

If "two" is used 9 times, then "nine" is used 7 times, so "seven" is the final word used more than once. One of "six" and "seven" is used 2 times while the other is used 3 times. But each of "five" and "three" is used 6 or 7 times, so our count will be too high, contradiction.

---

We're still assuming $n\leq23$, so now $n=23$ is the only case left. Here "times" is used 24 times, "time" is used 109560 times, and so "one" is used 109564 times. Now "four" is used at least 3 times, so it becomes the 21st word on our list. We know "two" is used at least 8 times; let's say it's used 9 times, so that "nine" is used 7 times and "seven" is the 22nd word on our list. After a little trial-and-error and deduction, I got to the following list:

• "this" ($2$ times)
• "sentence" ($2$ times)
• "uses" ($2$ times)
• "or" ($2$ times)
• "mentions" ($2$ times)
• "and" ($2$ times)
• "the" ($109583$ times)
• "word" ($109583$ times)
• "times" ($24$ times)
• "time" ($109560$ times)
• "one" ($109564$ times)
• "hundred" ($9$ times)
• "thousand" ($5$ times)
• "eighty" ($3$ times)
• "nine" ($7$ times)
• "five" ($6$ times)
• "two" ($9$ times)
• "three" ($7$ times)
• "twenty" ($2$ times)
• "sixty" ($3$ times)
• "four" ($3$ times)
• "seven" ($3$ times)
• "six" ($2$ times)

... which all works out, giving the solution I put at the top. Hurray! :-D