MathDB

2

Part of 2017 Federal Competition For Advanced Students, P2

Problems(1)

necklace with 2016 pearls, black, green or blue

Source: 48th Austrian Mathematical Olympiad National Competition (Final Round, part 2) 24th May 2017 p2

5/25/2019
A necklace contains 20162016 pearls, each of which has one of the colours black, green or blue. In each step we replace simultaneously each pearl with a new pearl, where the colour of the new pearl is determined as follows: If the two original neighbours were of the same colour, the new pearl has their colour. If the neighbours had two different colours, the new pearl has the third colour. (a) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if half of the pearls were black and half of the pearls were green at the start? (b) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if thousand of the pearls were black at the start and the rest green? (c) Is it possible to transform a necklace that contains exactly two adjacent black pearls and 20142014 blue pearls to a necklace that contains one green pearl and 20152015 blue pearls? Proposed byTheresia Eisenkölbl
combinatoricsColoring