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3n + 1 is interesting because it seems so tantalizingly easy and beautiful at first glance. Its like looking at a stream where you can clearly see the stony bed and it appears to be ankle deep but once you take a step you realize the lighting has fooled you and you fall head over heels into icy waters.

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It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0.

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).

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