The Constitution indicates that in Spain the electoral districts are the provinces and the Autonomous Cities of Ceuta and Melilla. This division has a certain roots and is not manipulable, even if as a result of population variations, the number of deputies to be elected in each province changes, as will happen in the general elections of 2019 (Madrid and Barcelona will choose 37 and 32 respectively, when in 2016 they had chosen 36 and 31, while Valencia and Asturias went from 16 and 8 deputies to 15 and 7).
On the contrary, in countries where a single representative is elected in each district, the districts must be redesigned each time the population varies. In the United States, for example, districts are drawn every 10 years, when a new census is taken. Although it is done taking into account geography (which, to simplify, it will be geometry for us), this gives great power to the organ responsible for making the design, and can lead to manipulations and the phenomenon known as Gerrymandering, of which this same newspaper has shown the following simple example.
Imagine a state composed of 25 people who must choose five representatives between two parties: green and blue. 60% of the population votes green and 40% blue. In the first image, the outline of the five electoral districts makes the green party get three representatives and the blue party two, with the number of representatives of each party proportional to the vote. However, in the second design the green party takes the five representatives and in the third example only two, despite having the majority.
Our challenge, or challenges, because there are two, propose a similar situation, now with 225 voters, 120 favorable to the ocher party and 105 to the gray party (we chose these colors to avoid identifications with really existing parties). You have to choose 15 representatives, each of whom will represent the 15 electors of a district. The only requirement for designing the districts is that the 15 electors of each of them have to form a continuum. That is, a district can not have 2 pieces, although we admit that 2 voters touch only one corner. The one in charge of designing the districts is an evil Governor, who belongs to the gray party, and therefore he is interested in that the grays obtain the greater number of possible seats.
What is that maximum number of gray seats if the voters are arranged as in this table?
What if the distribution was this?
In each of the two cases you have to indicate a map of districts with which the grays get the representatives that are said, and explain rigorously why you can not draw a map with which you get more.
We have hidden the comments in this news to avoid publishing the solution.