MathDB

Let

n,k

be positive integers such that

n\ge k

n

lamps are placed on a circle, which are all off. In any step we can change the state of

k

consecutive lamps. In the following three cases, how many states of lamps are there in all

2^n

possible states that can be obtained from the initial state by a certain series of operations? i)

k

is a prime number greater than

2

; ii)

k

is odd; iii)

k

is even.

Problem Statement