Let 2001 given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point P have the same colour as P, then the colour of P remains unchanged, otherwise P obtains the other colour. Starting with the first colouring F1, we obtain the colourings F2,F3,…. after several recolouring steps. Prove that there is a number n0≤1000 such that Fn0=Fn0+2. Is the assertion also true if 1000 is replaced by 999? combinatorics proposedcombinatorics