Can You Solve The Dice Rolling Drinking Game?

The game starts with 6 empty glasses in a row numbered 1 to 6. You roll a standard die. If the number for the glass is empty, then the glass is filled up. If the number for the glass is full, then you drink that glass. There is a special rule when 5 glasses are full. If you roll the number for the lone empty glass, then the final glass gets filled and you have to drink all 6 glasses. At this point the game ends.

From the start of the game, what is the average number of rolls until the game ends?

Blog post (text explanation): http://wp.me/p6aMk-4tu

I read about this game on Reddit Math: https://www.reddit.com/r/math/comments/408a45/how_many_turns_does_it_take_on_average_until_this/

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