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        riddles >> medium >> AYANS (and yet another number sequence)
        
(Message started by: NickH on Jun 12^th, 2003, 1:50pm)

Title: AYANS (and yet another number sequence)
Post by NickH on Jun 12^th, 2003, 1:50pm

Complete the sequence:

1, 21, 19, 18, 20, 23, 4, 8, 13, 2, 11, 5, 17, 16, 3, 10, 7, 12, 9...

Title: Re: AYANS (and yet another number sequence)
Post by NickH on Jun 14^th, 2003, 4:49pm

I've added four more terms to this sequence...

Title: Re: AYANS (and yet another number sequence)
Post by NickH on Jun 25^th, 2003, 2:26pm

I've added a further four terms...

Title: Re: AYANS (and yet another number sequence)
Post by wowbagger on Jun 26^th, 2003, 4:03am

I don't think adding more terms is of much help. That would be the case if one has a few rules, all producing the same initial sequence. On the other hand, if you can't get the first few elements of the sequence, adding information to the end won't help you.

May I ask whether your sequence has anything to do with words/letters? Or does it hail from a purely mathematical realm?

Title: Re: AYANS (and yet another number sequence)
Post by Sir Col on Jun 26^th, 2003, 10:10am

I was thinking the same when I saw that four more terms had been added. :D

Perhaps we need to pool together thoughts and observations, who knows we may inspire each other...?

It looks to me that there is, at least, a connection with letters. We can see that no number, so far, has repeated and none of the values exceed 26.

The alphabetical equivalent of,
1, 21, 19, 18, 20, 23, 4, 8, 13, 2, 11, 5, 17, 16, 3, 10, 7, 12, 9...
would be,
A, U, R, S, T, W, D, H, M, B, K, E, Q, P, C, J, G, L, I, ...

(i) I was wondering about writing the alphabet in a grid according to some rule, and then reading off the alphabetical position of letters in rows or columns so as to produce the sequence. However, we'd expect to find some consecutive letters, even if it spiralled.

(ii) There may be some type of modular arithmetic being applied, according to the alphabetical values: looping 1-26, e.g. 16+20=36==10.
If that were the case, the following partial string represent what is being added to each value to produce the next term.
After 1: +20,24,25,28,29,7/33,4/30,5/31,15/41,...

Hmm? Has anyone made any other observations? I'm sure that nothing is too trival, in that it may be the missing piece in someone else's jigsaw.

Title: Re: AYANS (and yet another number sequence)
Post by visitor on Jun 26^th, 2003, 10:53am

Some trivial meandering:
It's an ordered sequence, certainly not a straightforward mathematical progression. The letter order does not look very promising. Maybe it's something else reordered (such as the periodic table rearranged alphabetically or by year of the element's discovery, but that's not it). Think of some ordered list with somewhere between 23 and 26 or 27 elements.
The fact that it starts with 1 is intriguing, unlikely in a typical ordered list that's not mathematical, so what if it's a list that must be built one element at a time that would preserve the one's head location: he could have started with 1, then 1,2, then 1,2,3, then 1,4,2,3, then 1,4,2,5,3, then 1,4,2,5,3,6. But I can't imagine why he would have intended from the start to end the list at 23-26. This method could also work for letters, if there was some reason the D would belong right after the A (from which it got separated by the addition of later letters).
Maybe it's worthwhile to consider the elements that are missing: 6,14,15,22 (and possible 24,25,26). If 23 is the highest number, there are only 24 possible answers to complete the sequence. we each guess a different sequence and see who Nick picks. I'll guess 14,6,15,22. Do I win?

Title: Re: AYANS (and yet another number sequence)
Post by wowbagger on Jun 26^th, 2003, 11:23am

on 06/26/03 at 10:53:59, visitor wrote:

we each guess a different sequence and see who Nick picks. I'll guess 14,6,15,22. Do I win?

;D

I'll pick 22, 15, 6, 14.

Title: Re: AYANS (and yet another number sequence)
Post by NickH on Jun 26^th, 2003, 2:44pm

Quote:

May I ask whether your sequence has anything to do with words/letters? Or does it hail from a purely mathematical realm?

It is purely mathematical. As a clue, arrange the numbers <added>1 to 23</added> in a ring.

Title: Re: AYANS (and yet another number sequence)
Post by SWF on Jul 6^th, 2003, 9:16pm

My guess is 6, 15, 14, 22 and there is a good reason:
[hide]
Start with the numbers 1, 2, ... 23 arranged clockwise in a circle. 1 is the first number selected, erase it. Going clockwise around the circle, step through the remaining numbers one at a time until the 42nd step. That arrives at 21, the next number of sequence. Erase the 21. Now step 41 steps through the remaining numbers and arrive at 19. 40 steps gives 18; 39 steps gives 20; 38 steps gives 23, ... Reduce the number of steps by 1 each time, until: 24 steps gives 6, ... [/hide]

Title: Re: AYANS (and yet another number sequence)
Post by NickH on Jul 9^th, 2003, 1:06pm

SWF, that is the answer I was looking for, and very nearly the technique I used to generate the sequence. Your method is equivalent to always taking exactly 20 steps through the numbers!

The problem of determining the last number in such a sequence is known as the Josephus Problem (http://mathworld.wolfram.com/JosephusProblem.html).