Two parked cars creeping forward to spaces 3 and 4

A row of four parking spaces, numbered $1$ to $4$, has cars in spaces $1$ and $2$ while $3$ and $4$ are empty. Each minute, a car may move ahead by one space only if (i) that space was empty at the start of the minute and (ii) the car is not already in space $4$. Every car allowed to move does so with probability $\tfrac12$, independently. So in the first minute only the car in space $2$ is allowed to move. What is the expected number of minutes until the cars sit in spaces $3$ and $4$?

$20/3$. Solve a small Markov chain on the cars' positions. Getting from $(1,2)$ to $(1,3)$ takes $2$ minutes on average. From $(1,3)$ both cars may move, so the next minute leads to $(2,4)$, $(1,4)$, $(2,3)$ or stays put, each with probability $\tfrac14$; solving the expected-time equations from there gives $E_{13}=\tfrac{14}{3}$ and $E_{12}=\tfrac{20}{3}$.