Rolling until the running total is even

A fair die is rolled repeatedly, and you stop as soon as the running total of all rolls so far is even. What is the expected number of rolls?

$2$. Half the time the first roll is even and you stop at once. Otherwise the total is odd and you need another odd roll, which takes a geometric number of rolls with mean $2$. So $\mathbb{E}[N]=\tfrac12\cdot1+\tfrac12\cdot(1+2)=2$.