MathDB

Let

n

be an odd positive integer. Given are

n

balls - black and white, placed on a circle. For a integer

1\leq k \leq n-1

, call

f(k)

the number of balls, such that after shifting them with

k

positions clockwise, their color doesn't change. a) Prove that for all

n

, there is a

k

with

f(k) \geq \frac{n-1}{2}

. b) Prove that there are infinitely many

n

(and corresponding colorings for them) such that

f(k)\leq \frac{n-1}{2}

for all

k

Problem Statement