Lesson Three grasshoppers play leapfrog along a line

Three grasshoppers play leapfrog along a line

Problem 1

The initial configuration is like this:


-----0--------A------B-------------C----------


Three grasshoppers, named A, B and C, were sitting on the straight line (number line !).

Let's consider the function 


f = 


where , ,  are coordinates of the grasshoppers A, B and C at each current turn of the game.


The function f takes the values +1 or -1 depending on the position of A, B and C at each current moment. 

In the initial position the value of the function f is +1.

Notice that f changes the sign to the opposite at each and every turn of the play.

Then after 1991-th turn f will have the opposite (negative) sign to that "+1" it had initially.

It means that the grasshoppers can not return to (can not be at) their initial positions after 1991-th turn.