Is it possible to construct a sequence that ends in 1000000000?

Question

Starting from the number $1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it or rearranging its digits (not allowing the first digit of the rearranged number to be $0$) for instance a sequence might begin $$1,2,4,8,16,61,122,212,424,\ldots$$ Is it possible to make a sequence that ends in $1000000000$ and a sequence that ends in $9876543210$. Please show me how and if there is any working.

Answer 1

We first note that we can attain $1000$ from $1$ by $$1,2,4,8,16,32,64,128,256,512,125,250,500,1000$$ Therefore $1000^n$ can be attained by performing the same operations as above with $1$ replaced by $1000^{n-1}$.

On the other hand, note that rearranging the digits does not change the remainder when divided by $3$, and $2^n \neq 0 \pmod{3}$. Therefore the sequence would not reach any multiple of $3$.