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Tangled up in Collatz
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
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
no subject
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.
no subject
If you want to play around with it yourself:
http://skanderkort.com/collatz_conjecture_calculator