Draws needed to clear a deck of ten pairs

A deck has $20$ cards: two copies of each value $1$ to $10$. Each turn you draw two cards at random from what is left. If they show the same value they are discarded for good; otherwise both go back. The game ends when the deck is empty. What is the expected number of turns?

$100$. With $j$ pairs left, a turn removes a pair with probability $\tfrac{1}{2j-1}$ (the second card must match the first), so that stage lasts $2j-1$ turns on average. Summing over $j=1,\dots,10$ gives $1+3+\dots+19=100$.