(After reading the text below, scroll down for a visual explanation of a watershed analogy.)
For now, the two ways are
1) Between five to thirty years after the school opens:
The attendance of their child at the school will cause politically connected parents to experience cognitive dissonance with their attitude to the other political side: They might dislike and distrust the other goverment or political structure, but now their children are going to school with children from this other group and partaking in cooperative learning. Assuming they keep their child enrolled, the parents' attitudes towards the other side will improve as they witness more and more of their child's academic success in these group encounters at school.
Over the years, with this happening to enough politically influential parents, their colleagues and superiors will also experience cognitive dissonance: This second group - those without children at the school - might not trust the other side, but now they have colleagues whose attitudes have changed, and now there is this school that is systematically trying to enroll the most politically connected children each year, with plans for it to continue "ad infinitum." This second group will thus also feel the pressure to improve their trust and attitude towards the other side. Eventually this cognitive dissonance will flow up to the leaders themselves, especially once they themselves have their own children or grandchildren attending the school.
This process is very slow and imperceptible.
Increased trust among the politicians will lead them to cooperate even more than now during joint problem-solving sessions about the problem, and more cooperation during problem solving leads to more creativity (in finding a solution). The more you and I trust and feel comfortable with each other, the more we can share our crazier ideas, one of which might actually be the solution.
Five to thirty years might seem like a long time, but all children have to go to school somewhere, every year. See the watershed analogy below.
2) Looking beyond the first 30 years of the school:
No one knows who the politicians will be in the long term, but the people most likely to have a certain job in the future are the children and other young family members of the people who currently have that job. There will be a good chance that the alumni of the school - who will have learned how to cooperate with the other side in order to excel in their academic studies - will end up as government employees for their jobs. Graduating from the school, however, would not guarantee employment in the government or in any other job.
The more the alumni of the school become the politicians and other government officials in the decades ahead, the larger the critical mass there will be of people holding political influence who identity first and foremost with the social trait that unites the two sides. The more this critical mass can find its way into the higher circles of power, the larger role they can have in setting up creative joint-brainstorming sessions about the problem, such as having each individual argue aloud for both of the opposing perspectives. This last method can lead to a lot of creativity.
Also in the long term, the general populace, through what sociologists call "weak ties," will know that the children of their side's leadership are going to a school with the children from the other side's leadership, a school where cooperation is systematically structured into the lessons of all subjects in all grades. This awareness will gradually build their trust of the others.
How many students over how many years have to attend The Cypriot School to create enough positive pressure "downstream" for those at the negotiating table to think creatively enough to get a solution?