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?