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The 6666 Problem

Published online by Cambridge University Press:  23 January 2015

Roger Eggleton  and
Krzysztof Ostaszewski
Roger Eggleton
Affiliation:
Mathematics Department, Illinois State University, Normal, IL 61790, USA, e-mail: roger@ilstu.edu
Krzysztof Ostaszewski
Affiliation:
Mathematics Department, Illinois State University, Normal, IL 61790, USA, e-mail: roger@ilstu.edu
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Extract

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

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Articles
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The Mathematical Gazette , Volume 96 , Issue 535 , March 2012 , pp. 39 - 48
Copyright
Copyright © The Mathematical Association 2012

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