Let's say I have a regular fair die, but instead of digits 1 through 6, there are three 0s and three 1s.
I can generate a random $n$-digit binary number as follows: if initially I roll a zero (or a series of them), I ignore them; then what is left to do is just roll the die $n$ times. [Thanks to user Alex Meiburg, I have learned that this strategy does generate random numbers, but not uniformly; as he pointed out, generating, e.g., '101' is 3 (!) times more likely than generating '111'. Given this, I would appreciate if someone proposed a strategy lacking this oddity.]
Question: What is the simplest algorithm to generate a random $n$-digit number in base-$b$, $b\neq2$ using this "binary" die?
I have been trying to come up with an answer for a few hours, but I feel that there is none.