Question 21
On a necklace, $2016$ beads are arranged in a circle, each of which has one of the colours black, blue or green. In each step, every bead is simultaneously replaced by a new bead, where the colour of the new bead is determined as follows: if the two original neighbours had the same colour, the new bead has their colour. If the neighbours had two different colours, the new bead has the third colour.
a) Is there such a necklace, half of whose beads are black and the other half green, from which one can obtain a necklace consisting only of blue beads by such steps?
b) Is there such a necklace, on which a thousand beads are black and the remaining ones green, from which one can obtain a necklace consisting only of blue beads by such steps?
c) Is it possible, starting from a necklace that contains exactly two adjacent black beads and otherwise only blue beads, to arrive by such steps at a necklace that contains exactly one green bead and otherwise only blue beads?
a) Is there such a necklace, half of whose beads are black and the other half green, from which one can obtain a necklace consisting only of blue beads by such steps?
b) Is there such a necklace, on which a thousand beads are black and the remaining ones green, from which one can obtain a necklace consisting only of blue beads by such steps?
c) Is it possible, starting from a necklace that contains exactly two adjacent black beads and otherwise only blue beads, to arrive by such steps at a necklace that contains exactly one green bead and otherwise only blue beads?