Did Paul Erdős leave $500 sitting somewhere for the person who shows that mathematics IS ready for such problems?
Reversing from 1 would simply give you formulas that reach 1 (if the number 1 itself is, in fact, part of your math). That wouldn’t tell you anything except which numbers go to 1. Granted, it might produce gaps and, if the gaps stayed open long enough that might provide a number worth analyzing. But I don’t that this is a particular efficient means of search.
If you’re simply looking for pathways, rather than numbers, then whether you go through the path forwards or backwards isn’t particularly meaningful. A reversed formula is equivalent to a forwards formula. All that we really care about is whether there’s a path that loops and doesn’t include the number 1. Where that occurs is inconsequential so anything which would root it to the number 1, or any point in the number line, should be less efficient than any method which simply doesn’t care.
Of course, all search mechanisms are potentially less efficient than finding some math that proves the case one way or the other.
If someone can demonstrate that:
3^h / 2^g x + (3^h - 1) / 2
Will never produce a whole number, except the number 1, for whole numbers of h and g >= 0 and x >= 1, then you can conclusively state that there are no loops in Collatz. In theory, it might still diverge, but there are no loops.
That might be a simple thing for a mathematician. It’s a thing that I’m intending to investigate as well since it’s a very efficient means of proving something.
You can always take the 2n branch, but you can’t always take the (n-1)/3 branch. If you’re at 5, for instance, (n-1)/3 would give you 4/3, which isn’t an integer. And even if you’re at 7, (n-1)/3 would give you 2, an integer, but you still can’t take that branch, because 2 is even: Under the forward Collatz rules, 2 goes to 1, not 7.