There is already solution for the first of the electoral mathematical challenges presented by EL PAÍS and the Royal Spanish Mathematical Society. Angélica Benito Sualdea and Adolfo Quirós Gracián, professors of the Autonomous University of Madrid, they proposed the challenge and now they give us the solution.
Recall that we had 225 voters, gray and ocher, who must choose 15 representatives representing each of 15 voters, and it was to see which district design was most beneficial to the gray party with two different geographical distributions of voters. The only requirement was that the districts did not have two pieces.
Since there are 15 voters in each district, eight are needed to win it. In the two situations we proposed, the grays have 105 voters and 105/8 = 13,125. Therefore, the maximum number of representatives that can be obtained in any case is 13.
But with the first distribution of voters, which was that of this image,
Any district that matches the top gray voters with the bottom voters has to go through eight yellow voters and, therefore, the yellow party would win that district.
Looking at only the 60 gray voters above, we have 60/8 = 7.5 and, therefore, you can get seven representatives in the top strip. With the 45 votes below you have 45/8 = 5,625 and the grays can not surpass the five representatives there. So, with that distribution, the most that grays can get are 12 representatives, and they get them, for example, like this:
The second distribution of voters was this:
Now the grays can reach the 13 representatives. For example, designing the districts:
130 solutions have been received within the established period, coming from Spain as well as from other European and American countries. 93% of them are correct and almost all are adequately justified, with some examples of specially well-drawn districts, either because of their symmetry, disposition or color. This is the case of what Ruth L., Rafael C. or Ángel P. send us.
There are also "family" solutions, such as the one found jointly by Xavier V. and his son, or those that have been drawn up, in this case separately, Claudia and César C.
Some readers, including Eva G. or Asier G., express their dismay at what is concluded from the challenge. It could reassure them the comment of Javier R., who points out that "in practice a governor can never know with certainty the decision that voters finally adopt".
The RSME has decided to select a reader among those who have resolved the challenge to send a copy of the book Mathematical challenges, which includes in expanded versions those that were originally published in EL PAÍS on the occasion of the centenary of society and that is part of the Mathematical Stimulus Library that publishes jointly with Editorial SM. The graceful has been Rodrigo V.
Whether you have given the answer or not, we hope that the challenge has been interesting for you. We will be satisfied if it has been fulfilled what Luis M. del P kindly tells us: "It is an excellent way to talk about elections without arguing (too much) and having fun".
If this has been the case, we'll wait for you in the next one, which will again deal with circumscriptions, but this time in relation to the D'Hondt method.