Count up with Zeckendorf's theorem!

For all your silly time-killing forum games.

Moderators: jestingrabbit, Moderators General, Prelates

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 3:31 am UTC

Wikipedia wrote:Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.


1 = 1
2 = 10
3 = 100
4 = 101
5 = 1000
6 = ...?



Just for reference, and those unfamiliar with the Fibonacci sequence who want to play, the first few Fibonacci numbers are:

1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711

That should do us just fine, for now... :twisted:
Last edited by Sean Quixote on Wed Sep 28, 2011 8:29 pm UTC, edited 1 time in total.

curtis95112
Posts: 638
Joined: Thu Jan 27, 2011 5:23 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby curtis95112 » Wed Sep 14, 2011 2:25 pm UTC

Since you posted up to 5, I'll start with 6.

6 = 111
Mighty Jalapeno wrote:
Tyndmyr wrote:
Роберт wrote:Sure, but at least they hit the intended target that time.

Well, if you shoot enough people, you're bound to get the right one eventually.

Thats the best description of the USA ever.

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 3:00 pm UTC

7 = 1010, if I'm understanding this correctly.

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 3:27 pm UTC

Looks like 6 should have been 1001 since the Fibonacci numbers cannot be consecutive (i.e. your number should never have 2 ones next to each other).

8 = 10000

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 5:48 pm UTC

9 = 10001

That's right, a-wan. Basically the way I see it, what's going to go on here is we're gonna pretend like we're translating Zeckendorf's representation into a base system that resembles binary, in that it will contain only ones and zeros. The Fibonacci numbers will be our "orders of magnitude" or place values: from the rightmost digit, just go in your head, "1, 2, 3, 5, 8, 13, 21, 34, 55, etc..."

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 6:54 pm UTC

10 = 10010

This doesn't look like a very efficient number system for small numbers. I'm sure it gets better for larger ones.

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:08 pm UTC

11 = 10100

Yeah, not really. Never even though of that before, but yeah, not really. :P I guess, every time we reach another Fibonacci number, the efficiency ratio will improve by approximately a factor of phi?

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:25 pm UTC

12 = 10101

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:28 pm UTC

13 = 100000

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:31 pm UTC

14 = 100001

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:32 pm UTC

15 = 100010

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:52 pm UTC

16 = 100100

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 8:04 pm UTC

17 = 100101

Ah, 17... Anyone wanna take a gander as to what it has in common with 72, 305, 1292, 5473, et cetera?

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 8:15 pm UTC

18 = 101000

At a guess and a look at a couple of them, it looks like they're all 100...101

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 8:23 pm UTC

19 = 101001

Eh.. maybe. I dunno actually, because the answer I was looking for technically has little if anything to do with the Zeckendorf's representation. It has more to do with another thing that I came up with, but I never was sure what I should call it...

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 9:00 pm UTC

20 = 101010

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 9:32 pm UTC

21 = 1000000

Sean Quixote wrote:Ah, 17... Anyone wanna take a gander as to what it has in common with 72, 305, 1292, 5473, et cetera?

If you don't want a spoiler alert: You might not want to read my thread over in the math forum. ;)

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 9:57 pm UTC

22 = 1000001

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 10:00 pm UTC

23 = 1000010

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 10:07 pm UTC

24 = 1000100

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 10:12 pm UTC

25 = 1000101

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Thu Sep 15, 2011 3:53 am UTC

26 = 1001000

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Thu Sep 15, 2011 1:15 pm UTC

27 = 1001001

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Thu Sep 15, 2011 6:16 pm UTC

28 = 1001010

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Thu Sep 15, 2011 6:21 pm UTC

29 = 1010000

I've never actually written this stuff out before (and now I guess I shouldn't have to ;)) so I just realized another thing that's going on here: if someone came along one day and said, "I want to create a base system that only has two symbols (1 and 0), but let's say that numbers can only be written in such a way that the 1s never touch eachother..." This is also what you would come up with. :roll:

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 3:36 am UTC

30 = 1010001

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 3:55 am UTC

31 = 1010010

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 5:06 am UTC

32 = 1010100

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 2:05 pm UTC

33 = 1010101

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 3:09 pm UTC

34 = 10000000

User avatar
a-wan
Posts: 5
Joined: Thu Jun 09, 2011 3:51 pm UTC

Re: Count up with Zeckendorf's theorem!

Postby a-wan » Fri Sep 16, 2011 5:05 pm UTC

35 = 10000001

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 5:59 pm UTC

36 = 10000010

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 6:43 pm UTC

37 = 10000100

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 11:04 pm UTC

38 = 10000101

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Sat Sep 17, 2011 5:32 pm UTC

39 = 10001000

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sat Sep 17, 2011 8:50 pm UTC

40 = 10001001

Anonymously Famous
Posts: 240
Joined: Thu Nov 18, 2010 4:01 am UTC

Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Sat Sep 17, 2011 10:44 pm UTC

41 = 10001010

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sun Sep 18, 2011 1:19 am UTC

42 = 10010000

gaga654
Posts: 12
Joined: Tue Mar 01, 2011 1:37 am UTC

Re: Count up with Zeckendorf's theorem!

Postby gaga654 » Sun Sep 18, 2011 3:02 am UTC

43 = 10010001

User avatar
Sean Quixote
Posts: 229
Joined: Tue Sep 14, 2010 1:20 am UTC
Location: Ubeki-beki-beki-beki-stan-stan

Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sun Sep 18, 2011 3:23 am UTC

44 = 10010010


Return to “Forum Games”

Who is online

Users browsing this forum: Earthling on Mars and 4 guests