Chapter 8.1, Problem 34E

Exercises 33-37 deal with a variation of the Josephus problem described by Graham, Knuth, and Patashnik in [GrKnPa₉₄). This problem is based on an account by the historian Flavius Josephus, who was part of a band of 41 Jewish rebels trapped in a cave by the Romans during the Jewish-Roman war of the first century. The rebels preferred suicide to capture; they decided to form a circle and to repeatedly count off around the circle, killing every third rebel left alive. However, Josephus and another rebel did not want to be killed this way; they determined the positions where they should stand to be the last two rebels remaining alive. The variation we consider begins with n people, numbered 1 to n, standing around a circle. In each stage, every second person still left alive is eliminated until only one survives. We denote the number of the survivor by J(n).

34 Use the values you found in Exercise 33 to conjecture a formula for J(n). [Hint: Write $n=2^m+k$ , where m is a nonnegative integer and k is a nonnegative integer less than 2m]