Question 46
Pat has a pentagon, each of whose vertices is coloured either red or blue. Once an hour, Pat recolours the vertices as follows.
* Any vertex whose two neighbours were the same colour for the last hour, becomes blue for the next hour.
* Any vertex whose two neighbours were different colours for the last hour, becomes red for the next hour.
Show that there is at least one vertex which is blue after the first recolouring and remains blue for ever.
* Any vertex whose two neighbours were the same colour for the last hour, becomes blue for the next hour.
* Any vertex whose two neighbours were different colours for the last hour, becomes red for the next hour.
Show that there is at least one vertex which is blue after the first recolouring and remains blue for ever.