Get a number by doubling and re arranging Inspired by this meta code golf post
The question goes like this:
Starting with 1 you can repeatedly perform one of the following two operations:
Double the number 
or
Rearrange its digits in any way you want, except that there must not be any leading zeroes.
Given a positive integer $n$, determine the shortest possible sequence of integers to reach $n$ with the above process, if possible.
Now, the obvious way to accomplish this is brute force, but the language I am working with doesn't really allow it.
My question is, is there a mathematical approach to determine the sequence for any $n$ (If it exists) without brute force?
 A: I don't have a full solution. I don't even have a partial solution, but I have a (partial) strategy that I'd like to share with other folks working on this.
It seems to me that it's easier to work backwards. Clearly, the shortest path from 1 to $n$ is the shortest path from $n$ to 1 by following the inverse rules, i.e., either divide by 2 or rearrange the digits as in the original rules.
Given that, it seems that a good strategy - though I don't know that it's optimal (I don't think it is) - is to start by halving $n$ and keep halving $n$ until an odd number is reached (let's call it $m$). Then rearrange $m$'s digits to obtain an even number (if possible) smaller than $m$ (if possible - if more than one such number exists, select the smallest).
If it's not possible to rearrange $m$'s digits to obtain an even number (all of $m$'s digits are odd), then backtrack (to $2m$) and don't divide by 2 but, instead, rearrange to obtain an even number smaller than $2m$.
If it is possible to rearrange $m$'s digits to obtain an even number but it's not possible to do so and obtain a number smaller than $m$ then go with the rearranged-but-larger-than-$m$ value.
Yes, I realise that there are gaps in the strategy above (i.e., what to do when such and such situation happens). I haven't worked it out in detail yet. I was hoping to get the community's opinion on whether this is a strategy worth looking into more detail or if there's some obvious thing I'm missing that makes it a bad strategy.
The gist of the strategy is to divide by 2 as many times as possible. There may be some backtracks and there are values of $n$ for which there is no path to 1.
