A small prize will be awarded for the first correct solution sent to Tony Mann (A.Mann@gre.ac.uk) by a Greenwich undergraduate.
In the small country of Mathsland, the citizens are obsessed with politics. Each one passionately supports one of the three political parties, which are the Coffee, Milk and Water parties.
Whenever two citizens meet, they discuss politics. If they support the same party, they don’t change their allegiance, but if two citizens who support different parties meet, they are both so persuasive that each of them abandons their previous allegiance and supports the third party. Thus if Milk and Water supporters meet, they both change to support Coffee.
The repressive laws of Mathsland forbid any gathering of more than two people so all political discussions are limited to the above.
If at any time all citizens support a single party, that party will declare a dictatorship and the other two parties will be abolished.
Initially there are 13 supporters of Coffee, 15 of Milk and 17 of Water. Find the shortest possible sequence of meetings which results in a dictatorship, or prove that under these conditions no party will ever command the support of every citizen.