Rolling until the running total is a multiple of six

A fair die is rolled repeatedly, and you stop as soon as the sum of all rolls so far is divisible by $6$. What is the expected number of rolls?

$6$. Whatever the current total is modulo $6$, exactly one face makes the new total divisible by $6$. So each roll stops the process with probability $\tfrac16$ independently of the past, and the number of rolls is geometric with mean $6$.