A quasi-pangrammatic number is a positive integer whose English name contains at least one copy of each letter used to write all of the positive integers that precede it.

123456789 ("one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine") is a quasi-pangrammatic number: its name contains all letters execept for b, c, j, k, p, q and z, and no other letters are needed to write integers from 1 to 123456788 in words.

The smallest quasi-pangrammatic number is trivially 1 ("one") because there are no positive integers before it.

What is the smallest quasi-pangrammatic integer after 1?

  • $\begingroup$ I guess you want the ENGLISH quasi-pangrammatic number? Although it would be interesting to see, which language provides the smallest number... Bonus question ;c) $\endgroup$
    – BmyGuest
    Jan 4 '16 at 14:56
  • 4
    $\begingroup$ @BmyGuest See here for the bonus question ;-) $\endgroup$
    – GOTO 0
    Jan 4 '16 at 15:07
  • $\begingroup$ nb. in British English number names contain additional uses of the word "and", like "one hundred and one". Not sure if that affects your answer. $\endgroup$ Jan 10 '16 at 9:29

Thanks to @dmg for the correction

I think the answer is

12,468 (twelve thousand four hundred sixty eight). It covers the letters: adefghilnorstuvwxy.

Why I think it is the answer?

I wrote down the letters that come in the English names of 1,2....19,20,30,40,50,60,70,80,90. I chose these numbers only because other numbers will just repeat. For example in sixty two, I have already considered 'sixty' and 'two' separately. Now the letters that I wrote were: ONETWHRFUIVSXGLY. I had to take 8 in my smallest number because it is the only number that has a G. 12 for L and W. 4 or 5 for F. I chose four because it is smaller. 6 for X.

  • 3
    $\begingroup$ The same set of letters is generated by 12468. Otherwise a perfect approach! $\endgroup$
    – dmg
    Jan 4 '16 at 10:44
  • $\begingroup$ @dmg Then i am afraid my answer is wrong. I forgot that 12 also have L in it. You should post it as answer. $\endgroup$
    – manshu
    Jan 4 '16 at 10:47
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    $\begingroup$ It's a minor mistake, your approach is correct, so I suggest you simply edit the number. $\endgroup$
    – dmg
    Jan 4 '16 at 10:48
  • $\begingroup$ @dmg Thanks for the approval. Corrected $\endgroup$
    – manshu
    Jan 4 '16 at 10:49

I wrote a little proramm to check the numbers from one to one million for this property and there turn out to be 817 such numbers.

I also checked it for German and due to äöüß there is no such number in the first million besides 1, so I replaced them with ae,oe,ue,ss and 721. With the smallest being 12,467, just one below the English one.

The smallest common number ($\neq 1$) in English and German is 412,678.

Here is a little plot of them:

enter image description here


In the meantime I also calculated the smallest German quasi-pangrammatic number using the correct spelling (with äöüß) and it turns out to be 4,512,637

  • $\begingroup$ That was the same number Manshu got - however, you failed to account for the fact that 12 is written "twelve". Nice work though! $\endgroup$
    – Deusovi
    Jan 7 '16 at 7:29
  • $\begingroup$ I think I don't understand your comment. Isn't 12,468 the right answer? $\endgroup$
    – jens_bo
    Jan 7 '16 at 9:35
  • $\begingroup$ You said the answer was 412678. $\endgroup$
    – Deusovi
    Jan 7 '16 at 9:44
  • 3
    $\begingroup$ Ah, sorry. I guess my text isn't clear enough. 412678 is the smallest quasi pangramatic number that exists in both English and German (with the write letters). Besides that I get manshu's result. :) $\endgroup$
    – jens_bo
    Jan 7 '16 at 10:06

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