A circle and six sectors

Question

A circle is divided into six sectors and six numbers 1,0,1,0,0,0 are written clockwise,one in each sector. in one step, we can add one to the numbers in any two adjacent sectors. Is it possible to make all the numbers equal? If so how many steps this can be achieved?

Answer 1

Hint: number the sectors from $1$ to $6$ around the circle. Think about the difference between the sum of the odd numbered sectors and the sum of the even numbered sectors.

Answer 2

Let us denote the 6 sectors by $ X_1, X_2, \cdots , X_6$, of which $1,0,1,0,0,0$ are the starting values, then we consider the target, which is to make all numbers equal by the operation specified.

When all the numbers become equal we should have $V = (X_1 - X_2) + (X_3 - X_4) + (X_5 - X_6) = 0$. The initial value of V is 2 and the operation of adding one to the numbers in any two adjacent sectors does not change this value of V. So, we can see that V is invariant and we can not make all the numbers equal.