ch3cooh: (Fringes of Chaos)
[personal profile] ch3cooh
These pictures are from the Elementary math circle program at SFSU.  It's a class I teach every Thursday with a coteacher, Josh R. :)  I have never taught with a coteacher this extensively before -- it's awesome!  One of the best parts is that he and I have very different mathematical backgrounds, so whereas the Graph Theory class I'm prepping for next week is foreign content for him, the past 3 weeks that we've spent studying sequences, building up the the Collatz conjecture is new content for me.

The ribbons the students (and instructors) are playing in are sequcens that start with any number of the students' choosing and then progress using the following rules:  If the latest number is even, divide it by 2 to find the next number.  If the latest number is odd, multiply it by three and then add one to find the next number.  The magic is that, so far, every initial choice of a number eventually leads to the same behavior - eventually, the sequence trickles down to the cycle 4-2-1.  But it's unproven if this is true for ALL numbers... it just works for all of the billions tested so far. :D  And sometimes it takes a while for the sequence to get down to that cycle.  For example, our TA chose to start with the number 231 which has a sequence that goes for 127 steps before cycling.


I want to try a similar session out with an older group and combine it with logic tables of pairity. :)  And I think this might be the last part of an "infinity" curriculum that I've been trying to piece together! :D

20141016_180015 20141016_181753 20141016_181547

Date: 2014-10-21 05:44 am (UTC)
From: [identity profile] greensword.livejournal.com
Looks like a lot of fun. :)

The magic is that, so far, every initial choice of a number eventually leads to the same behavior - eventually, the sequence trickles down to the cycle 4-2-1. But it's unproven if this is true for ALL numbers... it just works for all of the billions tested so far. :D

I'm vaguely astonished that this hasn't been proven. It just seems like the kind of thing that would have some sort of intuitive mathematical underpinning.

Date: 2014-10-22 05:47 am (UTC)
From: [identity profile] ch3cooh.livejournal.com
:) I agree - it's like a lot of problems that have nice bounds... but it also reminds me of chaos, and strange attractors in particular. I spent a while thinking about patterns in how a sequence following this pattern might increase in decrease. And then a while thinking about nearness to powers of two instead. And the 'feel' of both of those measures was very chaotic. Our TA chose 231 for example - which has a sequence that goes as low as 31 and as high as 3077 before hunkering down to the 4-2-1 cycle. I've added a picture of the sequence to the post.

If you want to play around with it yourself:
http://skanderkort.com/collatz_conjecture_calculator

Profile

ch3cooh: (Default)
ch3cooh

April 2015

S M T W T F S
    1234
567 891011
12131415161718
19202122232425
2627282930  

Most Popular Tags

Style Credit

Expand Cut Tags

No cut tags
Page generated Jul. 28th, 2017 08:51 am
Powered by Dreamwidth Studios