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?
Best Answer
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!