Suppose you start with the number $102$ and, at each step, you are allowed to double the number, add the number's digits, multiply its digits, or add 1. Your goal is to go from $102$ to $201$ in as few steps as possible. Of course, you could add $1$ $99$ times, but... where's the fun in that? Here is my best attempt:
$102 \to 3 \to 6 \to 12 \to 24 \to 25 \to 50 \to 100 \to 200 \to 201$
The short answer is:
No - there is not a shorter path
but
there are alternate paths. You can start the sequence with 102 doubled is 204 summed is 6...
or
Start with 102 doubled is 204 doubled is 408 summed is 12...
Your attempt uses the least amount of steps possible. I have written a computer program that calculates the shortest possible paths between $102$ and $201$ and it has found four with nine steps. I believe that these are all of them:
\begin{align*}102 \to 3 \to 6 \to 12 \to 24 \to 25 \to 50 \to 100 \to 200 \to 201\end{align*} \begin{align*} 102 \to 204 \to 6 \to 12 \to 24 \to 25 \to 50 \to 100 \to 200 \to 201 \end{align*} \begin{align*} 102 \to 204 \to 408 \to 12 \to 24 \to 25 \to 50 \to 100 \to 200 \to 201 \end{align*} \begin{align*} 102 \to 204 \to 408 \to 816 \to 48 \to 49 \to 50 \to 100 \to 200 \to 201 \end{align*}
I find the last one the most interesting.
