In the UK a political party is elected by earning a majority of seats. Each seat represents a geographical area (known as a constituency), all of which have different populations. This means that a government can find itself in the rather surprising position of having a majority of seats despite not having the majority of the votes.

Consider the simplified ‘country’ with 4 parties and 5 constituencies with populations 99, 199, 299, 399 and 499. Assuming 100% voter turnout, what is the minimum number of votes a party can obtain and still be elected? 

What about in a country with 10 parties and n constituencies, with populations 99, 199, … , 100n-1. What is the minimum number of votes a party can obtain and still be elected?