Rounds of two dice needed to see every face

Each round you throw two fair six-sided dice together and note both faces. On average, how many rounds are needed before every one of the values $1$ to $6$ has shown up at least once?

$70219/9240\approx7.60$. Track the number $i$ of distinct faces seen so far as a Markov chain. From $i$, a round adds $0$, $1$ or $2$ new faces with probabilities determined by $\tfrac{i}{6}$ and $\tfrac{6-i}{6}$; solving the expected-time equations backwards from $e_6=0$ (e.g. $e_5=\tfrac{36}{11}$, $e_4=\tfrac{271}{55}$) gives $e_0=\tfrac{70219}{9240}$.