Take a positive number. Let's give it a name: x. The rules of the game are this: (1) if x is even, replace it by x/2; and (2) if x is even, replace it by 3x+1. That's it, the complete rules of the game.
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This is a really fun thread. The 3x+1 problem is deceptively simple: the kind of thing anyone can understand & play with, but which the world's best mathematicians can't solve. A very brief explanation:https://twitter.com/AlexKontorovich/status/1172715174786228224 …
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If you play this game starting with x = 5 (say), then here's what happens. 5 is odd, so it gets replaced by 3x+1 = 16. That's even, so gets replaced by 16/2 = 8. That's even, so gets replaced by 8/2 = 4. You've likely gotten the hang of this now, & can see the end: 4 -> 2 -> 1
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Here's what the 3x+1 conjecture says: no matter what number you start with, eventually you'll end up at 1. It might take a very long time, but you'll get there.
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It's easy to write a 5-line Python program to check this up to, say, x = a trillion (or much further). But proving it turns out to be something else entirely! Lots of mathematicians have worked on this, and failed to solve it.
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It's a lovely problem, in part because you can goof around with it from first principles, & learn fun stuff. I did a lot of this when I was 18, 19 or so, & even today occasionally pull the problem out to relax. But people have also brought quite sophisticated mathematics...
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... to bear on the problem. Much has been learned, but the problem remains unsolved!
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More detail in the Wikipedia article: https://en.wikipedia.org/wiki/Collatz_conjecture … And, via
@geomblog, the news that Terry Tao has recently made some very interesting progress:https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ …Show this thread -
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Sigh.
@nathanju34 points out that in case (2) above I should have written "odd", not even, of course. My kingdom for an edit button.https://twitter.com/michael_nielsen/status/1172916056496558080 …Show this thread
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I was hoping he'd have weighed in on Terry Tao's new manuscript.
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Oh, I didn't know! Very interesting.
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