This sentence contains eleven words, twenty vowels and sixty one letters
This sentence contains twelve words, twenty one vowels and sixty five letters
(Verified both with online character counters)
While any number can be placed, there are limits as to what can go where. This greatly reduces the different numbers that need to be looked at.
- For the words, we have a lower bound of 10 (7 words with minimum 3 to be added) and a realistic upper bound of 12, as its unlikely any of the numbers will reach the hundreds.
- For the vowels, there is a lower bound of 16, (13 already and at least 1 vowel per word) and with words of numbers not having more than 4 vowels, we can set an upper bound of 26.
- For letters, we can set a realistic lower bound of 55 (41 and at least 5 letters per word to be added) and a rough upper bound of 70 ish, although this is the least accurate but also least important bound.
From here, it is simply trial and error for the words, but the vowels and letters can be done a bit more accurately. For 10 it becomes clear quite quick that there is no solution, as the letters part must be 1 word, i.e. 'sixty' or 'seventy' which is always too little or too large.
For 11 and 12, the vowels will be roughly around twenty. For 11, if the vowels is two words, the letters would have to be sixty or seventy, which doesn't work as the letter count is in the 61 to 67 range, so the vowels must be one word. With one word, the letter count is at 53, so it is evident that the word must be sixty ___, as 'fifty xxx' already is too much. With 'sixty' the letter count is 58, and it is obvious sixty one fits, and a quick check shows there are no other solutions.
For 12 we can adapt the previous answer. The number for vowels must now be two words, but it will still be low twenties, and the letter count will then increase by 3-5 so only 64 to 66 needs to be checked. Quickly checking 21, 22 and 23 shows that 21 is really the only possibility, and it is easy to then find sixty five.
And through this analysis, I can also be confident there are no other solutions.