The Collatz Trolley Problem

I enjoy a good trolley problem (meme). I came across this one and it presents an odd problem:

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All initial values of n thus far tested end up looping with 4, 2, 1, so if it’s any of those, I’m not sure how many people are sucked into the black hole. (Though it’s fewer than 5, so if the goal is minimization of deaths, pulling the lever is ideal regardless.)

This one is a bit odd to think about. On the one hand, at least 5×2^60 initial values have been shown to result in that loop. But many, many more have not (infinitely many, if you believe in infinities).

And if you look at the odd numbers in any sequence the geometric mean of the ratios of outcomes is 3/4, though this only means no divergence. Maybe there’s some cycle involving numbers bigger than 5×2^60.

Also it’s apparently been shown that for any m, the number of option for n between 1 and m is at least proportional to m^.84.

So on the one hand my gut says pull because that evidence sounds kinda compelling. But then some part of me recognizes that m^.84 isn’t even half of m for most m, and 2^61 is relatively small. But then there seems to be some sort of abductive principle allowing the practical inference that pulling is probably right, but I can’t tell what it is.

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