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What are the next three numbers in this sequence, and why?

110, 150, 230, 350, 550, 910, 1510, 2510...

Hint 1: here's another sequence that follows identical rules.

130, 210, 330, 530, 850, 1410, 2330, 3850...

Hint 2: here's a third sequence that follows identical rules.

30, 30, 30, 30, 30, 30, 30, 30...

Hint 3:

If you're looking for some complex mathematical formula, you're going down the wrong path.

Hint 4:

The solution requires some real-world knowledge, but it's very common knowledge.

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    $\begingroup$ Are you sure that 1510 isn't supposed to be 1430? $\endgroup$
    – shoover
    Commented Nov 7, 2019 at 4:30
  • $\begingroup$ @shoover yes, I'm sure. I just double-checked. $\endgroup$
    – TheSoundDefense
    Commented Nov 7, 2019 at 4:31
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    $\begingroup$ Well, I was able to find a pattern for the first six of each, using rot13(svobanppv pybpx gvzr qvssreraprf), but it falls apart in the seventh element in the first two sequences. $\endgroup$
    – shoover
    Commented Nov 7, 2019 at 4:45
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    $\begingroup$ Will the puzzle still be valid if we divide each number by $10$? $\endgroup$
    – WhatsUp
    Commented Nov 9, 2019 at 13:09
  • $\begingroup$ @WhatsUp no, the puzzle would not work in that case. The first two sequences would no longer be valid. (The third would, though.) $\endgroup$
    – TheSoundDefense
    Commented Nov 9, 2019 at 17:19

2 Answers 2

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Are the next 3 terms

110, 150, 230 ? (or 2510, 4150, 6910, 11510 if not wrapping, same logic as below)

The numbers are

times, written without a separator. To get the next term in the sequence, add (the hour multiplied by 40) minutes.

So:

1:10 + 1*40 = 1:50 + 1*40 = 2:30 + 2*40 = 3:50 + 3*40 = 5:50 + 5*40 = 9:10 + 9*40 = 15:10.

Continuing the sequence:

15:10 + 15*40 = 25:10 = 1:10 and we're back to the start

The same pattern applies for the second hint, and the third

is trivial as 00:30 + 0*40 will always be 00:30

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  • $\begingroup$ The wrapping around of the numbers was unexpected, but looking at where I stopped writing additional numbers in the sequence, I suppose it makes sense. I added one additional number to each sequence, to clarify that they should not wrap around. Otherwise you pretty much have it, though your reasoning is slightly more complicated than mine. $\endgroup$
    – TheSoundDefense
    Commented Nov 12, 2019 at 17:07
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    $\begingroup$ I thought it was deliberate! I've updated the next terms based on the change $\endgroup$
    – Mohirl
    Commented Nov 13, 2019 at 10:32
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    $\begingroup$ As long as 2510 was not added to the sequence Mohirl`s answer not only made sense but was more logical than the intended answer. (I think) $\endgroup$
    – balazs.com
    Commented Nov 13, 2019 at 10:45
  • $\begingroup$ Your new answer is correct! I've added another answer as well, to explain my approach. $\endgroup$
    – TheSoundDefense
    Commented Nov 13, 2019 at 17:05
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Mohiri got the correct answer, but they came to that answer for a slightly different reason than the logic I was using. I'm adding this answer to clarify my approach.

The next three numbers are 4150, 6910, 11510. The sequence is predicated on continually converting "minutes" into "hours and minutes". 110 minutes is equivalent to 1 hour and 50 minutes, or 1:50; thus, 150 is the next term. 150 minutes is equivalent to 2:30, and 230 is the next term; 230 minutes is equivalent to 3:50, making 350 the next term, and so on. This is why I clarified that the numbers did not wrap around after they increased beyond 2400, as 1510 minutes is not equivalent to 1 hour and 10 minutes.

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    $\begingroup$ Ha! That's definitely simpler $\endgroup$
    – Mohirl
    Commented Nov 13, 2019 at 17:08

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