# Is this sequence that hard?

I published a sequence that seems difficult to the general public, so I've thought of posting it here. It has a very simple logic (language independent). Can you figure out the next number?

• Would I sound pedantic if I said that there is a polynomial solution to any finite number sequence? And that there are always an infinite number of solutions anyways? – fffred May 18 '16 at 16:00
• I think the polynomial interpolation for these 9 numbers can be fairly complex compared to the intended solution. – Edgar G. May 18 '16 at 16:32
• @fffred I'm surprised Mathematica doesn't have a "generate number sequence puzzle solution" built-in, then we could really annoy people who ask these. – OrangeDog May 18 '16 at 16:58
• @fffred. Totally agree, there are infinite solutions to this. I get particularly irked when these problems are framed such that you're a 'genius' if you can figure it out (not the case here, but it is inferred a wee bit). I cant find it in web searches, but I remember reading about Feynman or maybe Neumann (very different personalities!) who kept interrupting someones speech with solutions when they posed a similar 'problem' (with a singular solution expected :). – Lamar Latrell May 18 '16 at 22:04
• @LamarLatrell OEIS wouldn't provide the infinite solutions that exist, but does have the one OP had in mind: oeis.org/A090928 – OrangeDog May 19 '16 at 6:50

Probably way over thinking this:

18

There is a hint that:

The background of the image is the Milky Way Galaxy, a spiral galaxy.

Reason:

Start by writing the natural numbers out in a spiral, as follows:

  25 10→11→12→13
↑  ↑        ↓
24  9  2→ 3 14
↑  ↑  ↑  ↓  ↓
23  8  1  4 15
↑  ↑     ↓  ↓
22  7 ←6 ←5 16
↑           ↓
21←20←19←18←17
Now flip the numbers vertically:
  21 20→19→18→17
↑  ↑        ↓
22  7  6→ 5 16
↑  ↑  ↑  ↓  ↓
23  8  1  4 15
↑  ↑     ↓  ↓
24  9 ←2 ←3 14
↑           ↓
25←10←11←12←13
And horizontally:
  17 18→19→20→21
↑  ↑        ↓
16  5  6→ 7 22
↑  ↑  ↑  ↓  ↓
15  4  1  8 23
↑  ↑     ↓  ↓
14  3 ←2 ←9 24
↑           ↓
13←12←11←10←25

• What! I chosed the Milky Way image randomly! And, yes, it turns out it is a hint! – Edgar G. May 18 '16 at 16:29
• How do you even think about this so quickly :o ? – Fabich May 18 '16 at 16:30
• Uh... This is "a very simple logic"? :P – KeyboardWielder May 18 '16 at 18:57
• Rather than doing all the flipping - Just create the spiral and then follow a spiral rotated by 180. – Joel May 18 '16 at 19:20
• Even though the OP didn't mean for the background image to be a hint, your attention to detail remains very impressive. – Aiman Al-Eryani May 18 '16 at 20:38

0 because it's the only digit not used yet

• arrgh! Beat me by seconds... :-) I have a feeling that there's more to this though... – Ben May 18 '16 at 15:47
• @Ben - 66 seconds is probably too long to be characterized as "seconds". ;) – TTT May 18 '16 at 19:07
• Personally, I often regard anything more than 0 seconds as "merely seconds" As in "We made love for hours, yet she correctly told me It was 'merely seconds.'" – Chowzen May 18 '16 at 20:21

10

Because

if you place a mirror in the middle of the sequence (between the 9 and the 2), every number and its reflection would add up to 11 (and the "?" is opposite the 1).

• Quite interesting, but then, how would you continue the sequence? (I didn't say it was infinite, though) – Edgar G. May 19 '16 at 7:29
• I assumed it was finite when I saw that you didn't deny it when @fffred commented "there's polynomial solution to every finite sequence". – Aiman Al-Eryani May 19 '16 at 7:38
• @AimanAl-Eryani: The "finite sequence" that fffred refers to is the nine elements supplied in the puzzle: 1,6,7,8,9,2,3,4,5. The point (s)he was making is that there's always a polynomial p such that p(i) is the ith element of the sequence, and since a polynomial is defined for all real numbers, you can always use the polynomial to trivial extend the sequence to an infinite one. (In fact, there are infinitely many such polynomials, though only one of minimal degree.) – ruakh May 19 '16 at 22:54
• If you inspect the 16 numbers following those given in the puzzle, and then the 24 numbers after that, you'll see similar. :) – Jeff Bowman May 20 '16 at 2:41

OEIS says

18

(Beginning of sequence A090928: "Permutation of natural numbers arising from a spiral.")

It includes a comment which is much the same as the reason given by p.s.w.g., but

doesn't flip. It just reads out counterclockwise after writing clockwise:

 17.16.15.14.13
18..5..4..3.12
19..6..1..2.11
20..7..8..9.10
21.22.23.24.25

(Actually, that's one of 2 sequences in OEIS. There's also one where the nine digits appear thirty some positions into the sequence. Wouldn't make a good puzzle.)

• (1) I don't know what "OEIS" is; please spell it out and provide a link.  (2) Answers on Stack Exchange should be self-sufficient and free-standing.  Even in conjunction with p.s.w.g.'s answer, I don't understand this answer.  Please explain the logic. (3) Use spoiler markdown, not code block. – Peregrine Rook May 18 '16 at 22:12
• @PeregrineRook: OEIS is the Online Encyclopedia of Integer Sequences. – jwodder May 18 '16 at 23:53

10

And my reason is better than Aiman Al-Eryani's because there is no arbitrary mirror position.

1 , 2,3,4,5 , 6,7,8,9 , 10,11,12,13 , ...

6,7,8,9 , 10,11,12,13 , ...

(Just take one block at a time alternating between the sequences.)

• This is the best answer, to me. "Simple logic" is to make as little steps as possible, and constructing a spiral is NOT simple. – Pimgd May 20 '16 at 10:29
• @Pimgd: Exactly. You might be interested in reading about Kolmogorov complexity, which is one way to make "as little steps as possible" precise. For example if we fix the Turing-complete language as Javascript, my solution is expressed by the function f(n){return (n-2>>3<<2)+(n+2)%8+2;}. I challenge anyone to beat that (in Javascript of course)! – user21820 May 20 '16 at 14:30
• I came up with 10 for the same reason. I agree it's the simplest conclusion. – Tab Alleman May 20 '16 at 14:48

Well I got 2.

Series is: 1, 6, 7, 8, 9, 2, 3, 4, 5, x

Start from central pair:

9 - 2 = 7

8 - 3 = 5

7 - 4 = 3

6 - 5 = 1

therefore, 1 - x = -1

=> x = 2

Is it

0
Because it's the only digit not represented yet...

It is

10

Because

it starts out 1, leaps out by 5, then does a sequence of 4 numbers, then jumps down by 7, repeat. Adding 5 to 5 yields 10.

14

Solution:

Take the natural numbers and always swap tuples with a length of 4. Starting from

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...

the resulting sequence is

1, 6, 7, 8, 9, 2, 3, 4, 5, 14, 15, 16, 17, 10, 11, 12, 13, 22, ...