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What are the next few numbers in each of these integer sequences? Each of them has a clear and justifiable solution.

1, 2, 3, 5, 8, 9, 12, 20, 21, 22, ??, ??, ...

50, 49, 48, 46, 45, 44, 43, 38, ..., 25, 24, 23, ??, ??, ...

11, 12, 20, 30, 80, 90, 10000, ????

... 4, 5, 6, 8, 10, 40, 46, 60, 61, ??, ??, ...

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    $\begingroup$ Hint: Although all of these are number sequences, none of them have anything to do with the actual numerical properties of the numbers themselves whatsoever. $\endgroup$ – Joe Z. Apr 6 '15 at 9:33
  • $\begingroup$ ,..., means that all numbers in the interval are included? $\endgroup$ – leoll2 Apr 6 '15 at 10:44
  • $\begingroup$ No, it means that the numbers in the sequence between 38 and 25 were omitted, and you need to find the numbers that come after 23. $\endgroup$ – Joe Z. Apr 6 '15 at 10:45
  • $\begingroup$ Any hint for the 2nd sequence? $\endgroup$ – leoll2 Apr 12 '15 at 9:14
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Partial answer

1)

23,25

Ordinal numbers that aren't made juxtaposing the cardinal and "-th" suffix.
Credit of this answer to Xyuzhg

3)

Googol

Numbers that when written have exactly 6 letters. 10000=myriad.

4)

64, 80

Nonnegative integers that when written out in full in English have no repeated letter.

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    $\begingroup$ Your answer for sequence 4 is correct. $\endgroup$ – Joe Z. Apr 6 '15 at 9:43
  • $\begingroup$ @JoeZ. I added a solution for sequence 3. If wrong i'll delete it $\endgroup$ – leoll2 Apr 6 '15 at 10:29
  • $\begingroup$ The rule is correct, but you haven't given me the next number. $\endgroup$ – Joe Z. Apr 6 '15 at 10:29
  • $\begingroup$ Googol is too big or ok? $\endgroup$ – leoll2 Apr 6 '15 at 10:37
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    $\begingroup$ Googol is the correct answer. Good job. $\endgroup$ – Joe Z. Apr 6 '15 at 10:38
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Sequence 1:

23, 25 (twenty-third, twenty-fifth)

Reason:

Numbers that when written in the chronological ordinal series (first, second, third, etc.) are not the number appended with "th". ("Eighth" is "eight" + "h".)

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  • $\begingroup$ Our solutions for 1 agree everywhere.. but yours is simpler. $\endgroup$ – Ben Frankel Apr 6 '15 at 12:22
  • $\begingroup$ Funny - 0 would be accepted by yours, but not mine. I suppose there are linguistic connections to be made here? $\endgroup$ – Xyuzhg Apr 6 '15 at 12:31
  • $\begingroup$ Well, it's not surprising that 'th' isn't added in bare to a number ending in 't' or 'y'; "two" is 1 of 2 numbers ending in 'o' and by coincidence it becomes "second"; and I guess that the disappearing 'e' makes some sense. Somehow 'dth', 'nth', 'rth', 'xth' are all natural and easy to pronounce. The other number ending in 'o', "zero", happens to not become something special and so gets 'oth', which is also not a problem to speak. $\endgroup$ – Ben Frankel Apr 6 '15 at 13:10
  • $\begingroup$ @Xyuzhg: I didn't include zero because I started counting from 1. $\endgroup$ – Joe Z. Apr 6 '15 at 21:05
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The second sequence is:

21, 20

And the pattern is

the number of stars on the US flag in reverse order.

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1) All numbers when written out (googol would be ten duotrigintillion) end in one of these letters: denortxy (sorted alphabetically). Take only the even spots of "denortxy" to get "eoty", and now all numbers ending in a letter from "eoty" are in our list.

..., 23, 25, 28, 29, ...

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  • $\begingroup$ Twenty ends in "y". $\endgroup$ – Joe Z. Apr 6 '15 at 10:34
  • $\begingroup$ Right, aeiouy then. $\endgroup$ – Ben Frankel Apr 6 '15 at 10:35
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    $\begingroup$ Also, "eight" ends in "t". $\endgroup$ – Joe Z. Apr 6 '15 at 10:35
  • $\begingroup$ Oh okay nevermind then, I'll think about it. $\endgroup$ – Ben Frankel Apr 6 '15 at 10:36
  • $\begingroup$ "Numbers that end in aeiouy when spelled out normally or that can end in e when spelled out how they sound" works for everything.. $\endgroup$ – Ben Frankel Apr 6 '15 at 10:38
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Sequence 2:

The sequence repeats its interval at every 25
50, 49, 48, 46, 45, 44, 43, 38
25, 24, 23, 21, 20, 19, 18, 13

my closest guess anyways

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  • $\begingroup$ There are numbers between 23 and 38 in the sequence. $\endgroup$ – Joe Z. Jul 14 '16 at 19:48

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