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One tough puzzle to solve :

Find the next number in the sequence :

5,105,74,712,37,?

Options are :

a. 2008
b. 57
c. 507
d. 98
e. 44

What is the next number, and why?

Puzzle Setter : Myself.

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    $\begingroup$ I'm not sure why this question has received so many upvotes. As far as I can see, it's just another one of those questions where we have to spot something obscure which links the sequence. Like all of these questions, all of these answers can be justified one way or other - even by a polynomial. What's more, the 'number-theory' tag is misleading: "A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers." So our answers are going to be revolved around numbers since that's what we're expecting. $\endgroup$
    – Shuri2060
    Commented Aug 2, 2016 at 16:40
  • $\begingroup$ I don't see how slimeArmy's answer is any less justifiable than the one you had us looking for. $\endgroup$
    – Shuri2060
    Commented Aug 2, 2016 at 16:44
  • $\begingroup$ For example, I could say that the numbers in the sequence are the roots of this polynomial: $x^6 - 977x^5 + 213877x^4 - 18980531x^3 + 757923574x^2 - 12347718440x + 45032433600$ which would give me: e. 44 $\endgroup$
    – Shuri2060
    Commented Aug 2, 2016 at 16:48
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    $\begingroup$ QuestionAsker, Such a rule could be created for any sequence of numbers. I think that the point of the puzzle is to find a rule that is simpler (has a lower entropy) than the sequence and that lets us find the next item without using the options (it's a rule, not a criteria). $\endgroup$
    – user9771
    Commented Aug 2, 2016 at 17:18
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    $\begingroup$ Had there been no options, There would be a huge number of answers to this question. $\endgroup$
    – Sid
    Commented Aug 2, 2016 at 17:20

4 Answers 4

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I think the answer is

$D:98$

Because

If we write the numbers given out in English and count the characters used (including spaces) they all have a multiple of $4$ characters: $(4,20,12,24,12)$. Of the options only $D$ has a multiple of $4$ characters: $(22,11,22,\underline{12},10)$.

Another answer could be

$A:2008$

Because

If we write the numbers given out in English and remove spaces they are in alphabetical order, and only $2008$ would keep that order.

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  • $\begingroup$ Not sure how either fits the number-theory tag though :/ $\endgroup$
    – Jonathan Allan
    Commented Aug 2, 2016 at 17:25
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The answer:

C: 507

Because:

None of the numbers in the sequence begin with an even number, which takes out option A and E.
The numbers also alternate between less than 100 and greater than 100, so the next number in the sequence would be greater than 100, removing B and D leaving C as the remaining option.

I'm looking for a more mathematical reason still, but is what I've conjured up as of now.

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EDITED:

May be a long shot, but: c.507. If taking "0" as addition, like how we get 6 from 1+5, 5+7=12, which can be distributed on the two extra 7's to make 8 and 9, completing 0123456789. Cannot explain how I can distribute though.

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    $\begingroup$ Sorry bro, not the right one. This is not an appropriate solution. Try it out. $\endgroup$
    – Bharatwaaj Shankar
    Commented Aug 2, 2016 at 16:01
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Ok, it's time for the answer. If you keep revolving around numbers then you won't be able to find the answer. It's lexicographically arranged numbers. Thus the answer is:

A: 2008

Thanks people for trying it out.

Five
One hundred and five
...
...
...
Thirty seven
Two thousand and eight.

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    $\begingroup$ Man, you could have waited some more time. This puzzle has less than a hundred views and hence, the number of responses is less. $\endgroup$
    – Sid
    Commented Aug 2, 2016 at 16:26
  • $\begingroup$ Haha. Let me try to give more puzzles. :D Thanks all. $\endgroup$
    – Bharatwaaj Shankar
    Commented Aug 2, 2016 at 16:27
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    $\begingroup$ Shouldn't sequence puzzles have one and only one possible solution from N? $\endgroup$
    – Erbureth
    Commented Aug 2, 2016 at 16:49
  • $\begingroup$ Welcome to Puzzling.SE. Please refrain from adding answers to your own question too soon, or even adding hints when not asked for. $\endgroup$
    – ABcDexter
    Commented Aug 2, 2016 at 16:50
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    $\begingroup$ Doesn't seven hundred and twelve precede seventy-four lexicographically? $\endgroup$
    – YowE3K
    Commented Aug 3, 2016 at 2:15

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