MathDB

Let

k

be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance

k

is the original distance. Formally, we may choose

2

tails

A, B

that are spaced

n

and move

A

to the different side of

B

so the current distance is

kn

. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of

k

so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.

Problem Statement