Problem 92

Problem
of the
Week

This is the last Problem of the Week for the semester, but see below for an additional question to think about over the summer. Please join us again at the start of the Fall Semester for more thrilling episodes of The Bradley University Problem of the Week.

PROBLEM 92

The following problem was suggested by Jens Voss. Vielen Dank, Jens!

Two players take turns throwing a die. The first one to roll a five wins. Of course, the first player has an advantage, but just how big? What are the exact winning probabilities for the two players?

For deeper thinkers: What happens if the game is played by n players who sequentially take turns rolling the die?

In the April 2000 issue of the American Mathematical Monthly, vol 107, no. 4, pages 360-364, is an interesting short article in which the authors, Neil Calkin and Herbert S. Wilf, describe a function which "counts the rational numbers." This function, f, from the positive integers to itself, has the property that every positive rational number occurs once and only once in the sequence {f (n) / f (n+1)}, as n ranges over the non-negative integers.

The question to ponder is the following: What is the inverse function; that is, given a positive rational number r/s, in lowest terms, for which integer, n, is

f (n) / f (n+1) = r/s? You want an inverse function which is easy to compute, elegant, and, of course, correct.

Enjoy your summer!!

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