Lecture 3

Learning objectives

After this class, you should be able to:

1. Define the Josephus problem.
2. Derive the recurrence relation for the Josephus problem.
3. From the solution pattern for small values of n for the above problem, guess the general solution, and prove its correctness using induction.
4. Given a variant of the above problem, derive a recurrence for it, and determine a closed form solution to it.

Reading assignment

1. Sec 1.3, pages 8 - 11 (until, and including, the induction proof).

Exercises and review questions

• Questions on current lecture's material
  1. Exercises 1.7, 1.15.

• Questions on next lecture's material
  1. Solve for f(2m + k), m > 0, 0 < k < 2m, where f(n) is given by the following recurrence:
     f(1) = 1,
     
     f(2n) = 2f(n),
     
     f(2n+1) = 2f(n).