Prove that the solution to the Josephus problem is 1 for every n that is a power of 2.

For the Josephus problem,

a. compute J(n) for n = 1, 2,..., 15.

b. discern a pattern in the solutions for the first fifteen values of n and prove its general validity.

c. prove the validity of getting J(n) by a 1-bit cyclic shift left of the binary representation of n.