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Topic: Conway Sequence - Audioactive Decay (Read 2856 times) |
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william wu
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Conway Sequence - Audioactive Decay
« on: Dec 28th, 2002, 4:55am » |
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Tricky if you've never seen it before. At least I thought it was.
What row of numbers comes next?
1
11
21
1211
111221
312211
13112221
Note: Apparently mathematician John Conway spent much time studying this sequence.
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| « Last Edit: Dec 31st, 2002, 4:37pm by william wu » |
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BNC
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Re: Conway Sequence
« Reply #1 on: Dec 28th, 2002, 6:14am » |
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I have an idea. Not sure if it's legal, but I'm thinking along the lines of:
look at the previous line as a text stream, rather than numbers
If that's legal, the following line should be:
1113213211
Note: Read that as:
11-13-21-32-11
1 "ones" then 1 "threes" then 2 "ones", etc.
And I must say: even if it's not legal / original answer, I am happy with it -- it's nice and follows the wording of the riddle.
BNC
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| « Last Edit: Dec 28th, 2002, 6:16am by BNC » |
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redPEPPER
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Re: Conway Sequence
« Reply #3 on: Dec 30th, 2002, 6:15pm » |
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As an easy bonus, prove that you never encounter the figure 4 in the series.
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Cyrus
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Re: Conway Sequence
« Reply #4 on: Dec 31st, 2002, 11:01am » |
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Maybe I should change my name from Cyrus to Dummy because I can't seem to get any of these. I never figured out what the next row of numbers was until I read the explanation by BNC.
The only thing I quickly noticed was the simple pattern at the end of each line alternating 11, 21, 11, 21 and if I had just looked closer at that, i should have realized it was one 1, two 1's, one 1, two 1's .... and that would have given the rest of the puzzle away.
Sometimes when I do these puzzles I get too caught up looking at the patterns in the numbers to remember that the numbers could stand for other things like letters or words or other things. (I guess I just have a simple mind)
As for the bonus question why you never see the number 4. I just looked at what would have to happen for there to BE a number 4. That would mean the previous line would have (for example) 1111 which stands for [one 1, one 1] and if that happened you would just write 21 not 1111.
I don't know if that is right or makes ANY sense. But I'm tryin' here. Maybe I can get at least an EASY BONUS question right.
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william wu
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Re: Conway Sequence
« Reply #5 on: Dec 31st, 2002, 12:57pm » |
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I wouldn't be too discouraged ... these puzzles are all very hard. If it makes you feel any better, I didn't get it either until someone hinted to me that every row only depends on the immediately previous row. I guess knowledge hurts sometimes, because I kept trying to match the pattern to other rules I've seen before -- namely Pascal's triangle and linear recurrences. Pascal's triangle because many of the rows are terminated on both ends by "1"s and bulge in magnitude in the middle, resulting in me multiplying polynomials and staring at the coefficients. Linear recurrences because with many sequences you scale and add two terms in the past to get the current term, resulting in me combining past rows.
The bonus proof sounds good to me.
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| « Last Edit: Dec 31st, 2002, 2:04pm by william wu » |
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william wu
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Re: Conway Sequence
« Reply #6 on: Dec 31st, 2002, 3:40pm » |
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Out of curiosity, I wrote some code to plot how the row length increases, and determine the fraction of digits that are 1s, 2s, and 3s (does it converge to a uniform distribution? is there a clear argument for why it would or wouldn't?). I only went up to 35 rows because runtime quickly gets long in MATLAB, although I'm probably just using a poor algorithm.
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| « Last Edit: Dec 31st, 2002, 7:08pm by william wu » |
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william wu
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Re: Conway Sequence - Audioactive Decay
« Reply #7 on: Dec 31st, 2002, 4:34pm » |
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@#$%! I was searching online for what exactly Conway discovered about this sequence and I found some insanely interesting stuff! This sequence, officially dubbed the Audioactive Decay sequence, is related to Conway's Lost Cosmological Theorem: There exists an integer N such that every string decays in at most N days to a compound of common and transuranic elements. You'll have to read it yourself:
Mario Hilgemeier's spiffy pages on Audioactive Decay, with nice pictures
http://www.btinternet.com/~se16/mhi/Part1.htm
AMS paper titled Proof of Conway's Lost Cosmological Theorem by Ekhad and Zeilberger
[URL]http://www.ams.org/era/1997-03-11/S1079-6762-97-00026-7/S1079-6762-97-00 026-7.pdf[/URL]
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| « Last Edit: Dec 31st, 2002, 4:36pm by william wu » |
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redPEPPER
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Re: Conway Sequence
« Reply #8 on: Dec 31st, 2002, 5:57pm » |
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on Dec 31st, 2002, 11:01am, Cyrus wrote:As for the bonus question why you never see the number 4. I just looked at what would have to happen for there to BE a number 4. That would mean the previous line would have (for example) 1111 which stands for [one 1, one 1] and if that happened you would just write 21 not 1111.
I don't know if that is right or makes ANY sense. But I'm tryin' here. Maybe I can get at least an EASY BONUS question right. |
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That's pretty correct. Except that four ones in a row could also mean: a number of 1, one 1, one something. But that doesn't make the demonstration any harder. You just have to keep that possibility in mind.
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BNC
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Re: Conway Sequence
« Reply #9 on: Jan 1st, 2003, 3:03am » |
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on Dec 31st, 2002, 5:57pm, redPEPPER wrote:
That's pretty correct. Except that four ones in a row could also mean: a number of 1, one 1, one something. But that doesn't make the demonstration any harder. You just have to keep that possibility in mind.
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Of course, in that case n1-11-1m should be (n+1)1-1m
And, n would have to be 1 or 2 becuse 4 is not possible, and thus n+1<=3 thus n<=2: n=1 or n=2
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wowbagger
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Re: Conway Sequence - Audioactive Decay
« Reply #11 on: Jan 6th, 2003, 7:42am » |
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The following (similarly generated) sequence has a nice property:
1 11 21 1211 1231 131221 ...
Trust me, it doesn't take too many iterations to find out. Be a man (or woman, whichever you prefer  ) and use pencil and paper!
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Cyrus
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Re: Conway Sequence - Audioactive Decay
« Reply #12 on: Jan 6th, 2003, 10:11am » |
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That's a cool one Wowbagger.
I always like the pencil and paper approach.
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BNC
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Re: Conway Sequence - Audioactive Decay
« Reply #13 on: Jan 6th, 2003, 11:11am » |
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1 11 21 1211 1231 131221 ...
132231 232221 134211
I did that one in my head... hope it's not a violation of the rules...
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Cyrus
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Re: Conway Sequence - Audioactive Decay
« Reply #14 on: Jan 6th, 2003, 12:15pm » |
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I needed pencil and paper to carry the sequence out to the 15th number until I realized the really cool thing about this sequence.
I guess it pays never to stop a sequence like this after only a couple numbers.
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wowbagger
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Re: Conway Sequence - Audioactive Decay
« Reply #15 on: Jan 6th, 2003, 2:41pm » |
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BNC: Your head is working just fine, as far as I can tell. 
on Jan 6th, 2003, 12:15pm, Cyrus wrote:| I needed pencil and paper to carry the sequence out to the 15th number until I realized the really cool thing about this sequence. |
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Are you sure it's the 15th term? Or did you calculate that one just to make sure?
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Jeremiah Smith
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Re: Conway Sequence - Audioactive Decay
« Reply #16 on: Jan 6th, 2003, 4:01pm » |
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From what I can tell, starting the "Wowbagger Sequence", as it were, with 1, 2, 3, and 4 all end up in the same way - repeating 14233221.
It also seems that any digit x, where x is 5, 6, 7, 8, or 9, ends up repeating 1x14332231.
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| « Last Edit: Jan 6th, 2003, 4:02pm by Jeremiah Smith » |
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Cyrus
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Re: Conway Sequence - Audioactive Decay
« Reply #17 on: Jan 7th, 2003, 8:19am » |
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I actually didn't have my trusty paper nearby, so I couldn't remember if it was 14 or 15.
Are there any more similar sequences to this?? I suppose you could make up almost anything using the same basic idea.
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zeb
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Re: Conway Sequence - Audioactive Decay
« Reply #18 on: May 8th, 2003, 2:15pm » |
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Here is an iteresting variation, maybe you have seen it:
A
ANA
ANAANNANA
ANAANNANAANAANNANNANAANNANA
It gets long very quickly, but that should give you the idea.
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Mike Sell
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Re: Conway Sequence - Audioactive Decay
« Reply #19 on: May 25th, 2004, 10:39am » |
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 I just found a series of small patterns.. none of this theorem and word stuff. a simple on for the first row was just 11211311.... and it worked for the first 3 rows. then using what was previously mentioned for the last 2, 11,21,11,21,11,21....for the 3rd from last it worked the same way.. 22122... and if that worked, it worked the same way vertically for column five.. 22122.... It wasn't as complicated as it seems. I guess intelligence does hurt, eh wu?
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Chekov
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Re: Conway Sequence - Audioactive Decay
« Reply #20 on: Mar 6th, 2007, 10:21am » |
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1
11
21
1211
111221
312211
13112221
the next lines are easy to get. The riddle is about literally saying what the line above means:
the first number is at the top. its just a 1. nothin else.
The next line: the top is 1(one) 1
the next line say: there are 2 ones
the next line: you will get. there is 1(one) 2. and 1(one) 1.
literally saying what the end = meaning
1113213211
31131211131221
13211311123113112211
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