The title is a reformulation of a game that is played as follows:
Before you are six light bulbs (all start turned off). You repeatedly roll a fair six-sided die and each time you roll a number you flip the switch of the corresponding switch, turning it on if it's off, and off if it's on.
What is the expected number of rolls it will take until every light bulb is turned on?
Note: I am not just asking "What is the expected number of rolls until each number has been rolled?", as this question has already been asked and answered many times here.
Best Answer
Denote by $e_k$ $\>(0\leq k\leq 5)$ the expected number of additional moves when there are $k$ bulbs alight. One then has equations like $$e_5=1+{5\over6}e_4,\quad e_4=1+{1\over3}e_5+{2\over3}e_3,\quad\ldots,\quad e_0=1+1\,e_1\ .$$ Solving this system gives $e_0={416\over5}$.