I'm very curious about the Collatz conjecture, also known as the $3n+1$ problem, mainly due to its rather simple formulation and beautiful visualizations. After all, Erdős himself said that "mathematics may not be ready for such problems", so there's a certain mystery about it that I find quite alluring.

I would like to present the conjecture to high school students, since I believe such type of problems are illustrative of how simple ideas yield complex, yet amazing, mathematics, and that may kindle the love for maths and number theory in particular. However, I'd like to do so in an informal manner, if possible via a real life application of the Collatz conjecture, but so far I haven't been able to find or come up with any interesting examples that could potentially captivate a not so mathematically trained audience. Any ideas?

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    $\begingroup$ This might be a better question for the Math Educators sister site. $\endgroup$ – Xander Henderson Aug 25 '19 at 23:54
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    $\begingroup$ How about not telling them that it’s famous, and just introducing them to the algorithm through a few examples, then offering bonus points if anyone can find an example that doesn’t eventually reach 1. Too diabolical? $\endgroup$ – Joe Aug 25 '19 at 23:57
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    $\begingroup$ I've tried that before with someone, and it was fascinating to see the effort put into finding that one example. Even more amazing was that person's realization of the hardness of the problem, so maybe that's the real reward and so it's not that diabolical (well, maybe the 'bonus points' bit..) $\endgroup$ – sam wolfe Aug 26 '19 at 0:01
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    $\begingroup$ Afterward you can tell them that if they actually find a counterexample, they’ll get a lot more than bonus points ;-) $\endgroup$ – Joe Aug 26 '19 at 0:21
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    $\begingroup$ @Joe (the quoted stopping time is $112$, I guess it's a minor difference in how I was reckoning it in my algorithm). But I still couldn't solve the general problem despite cracking my nut on it. It was only later when I srarted using the Internet that I learned this was actually a famous unsolved problem. So, short version: this has actually been done. $\endgroup$ – Deepak Aug 26 '19 at 0:58

If I were a high-school math teacher I would give each student a different $n$, have them each perform the Collatz transformation, and then report to the group the final (astonishing) unanimity of results, despite the large variance in sequence times. I'd then show a computer-generated list of the sequence for some extremely high $n$ (e.g., $3 \times 10^9$). I'd then let students suggest another "near Collatz" algorithm (e.g., $5 n + 1$) and show that this can explode. There is something special about Collatz's specific formula... but nobody knows what it is!

I would then say that for over 80 years some of the best mathematicians have been unable to solve this, that it has been verified by computer up to $2^{66}$ (write that out in full for them), and then give the famous Erdos quote.

I'd also talk about how a Andy Wiles, age ten, came across a different easy-to-state math problem and solved it hundreds of years after Fermat, got on the front page of every newspaper in the world and won a million dollars. Tell them that perhaps—just perhaps—one of them will make history by solving Collatz's intriguing problem. (Tell them that you tried and failed.... that will motivate them!)

I would show the first dozen or so nodes of the Collatz tree graph... just enough that an industrious student with time and plenty of paper can extend it at home.

Forget trying to make it "practical." Stand up for the beauty and profundity of high-level math and be glad that even high-school students can understand this extraordinary problem!

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    $\begingroup$ +1 for "Forget trying to make it practical". The idea that math has to be sold on it's practicality is anathema to what math is really about. Sell math on it's fascination and its beauty, because that's the only thing that matters at the end of the day. $\endgroup$ – G Tony Jacobs Sep 4 '19 at 23:07
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    $\begingroup$ Well, I am $100\%$ in favor of selling higher math--and particularly the Collatz conjecture--on its beauty, profundity, astonishing variety, and as an illustration of the extraordinary creativity and hard work of mathematicians over millennia. However, as a PhD Physicist who uses advanced math, I don't think it is "the only thing that matters at the end of the day." $\endgroup$ – David G. Stork Sep 5 '19 at 0:57

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