At college, my friends and I would sometimes waste time playing a game with dice. We would roll 25 dice, pick out all the dice that landed on a 6, then roll the rest. This would carry on until all the dice were removed. The winner was the person who managed to remove all the dice in the least amount of turns. (A turn counts as rolling the dice - i.e. the first turn is rolling all 25 dice, the second turn is rolling all the dice which didn't roll six on the first turn, etc.)
I want to find the expected amount of turns it would take to finish the game, but I am having difficulty modelling it. I know that if there was one dice then you could model it using the geometric distribution; and the number of sixes on each turn can be modeled using the binomial distribution, but I do not know how to combine the two (or if this is even useful).
Please help, this problem has been bugging me at the back of my head for several years.