Drawing card pairs until the deck is cleared

A deck of 20 cards contains two cards of each value from 1 to 10. Each turn you draw two cards at random from what remains; if they have the same value they are discarded, otherwise both go back into the deck. You continue until the deck is empty. What is the expected number of turns?

$100$. With $k$ pairs left, a turn succeeds when the second card matches the first, probability $\tfrac{1}{2k-1}$, so the wait averages $2k-1$ turns. Summing over $k=1,\dots,10$ gives $\sum(2k-1)=10^2$.