The 6666 Problem

How many times must a die be thrown to get four consecutive sixes? More generally, and more precisely, Q1. What is the expected number of times we must throw a die to get n consecutive sixes? The answer, rather suggestively for the numerologically inclined, turns out to be A1. The expected number of throws is 6 + 62 + 63 + … + 6n. In some well-known dice games players throw several dice at the same time: in craps a pair (brace) of dice is thrown, while in yahtzee, yam and balut, up to five dice are thrown at a time. However, Q1 is not the same as