Playing with triangles

The workings

Three horses pull better than two. This obvious idea is one of the keys to the popularity of the troika, a Russian draught created in the 17th century that allowed a single person to control three horses at once, and of course, to travel at quite high speeds for the time. In Russian troika (тройка) means “trio”, and that word was also used to define the three-person collegiate governments that existed at some points in the Soviet Union (some with worse fortune than others), i.e. our well-known triumvirate. Perhaps because of this second idea, the consortium (really a “decision group”) formed by the European Central Bank, the European Commission and the International Monetary Fund, which recently took charge of controlling the financial bailouts and economic adjustment programmes of certain European Union countries, was called the troika. As this group will perhaps evoke unpleasant memories for some readers, we recommend listening to the eponymous movement from Prokofiev’s Lieutenant Kijé suite (in which we can also hear a triangle) to recover. And of course, be entertained by the divertimento below.

The fun

A figure consisting of an equilateral triangle and a line parallel to one side passing through its centre is given. A game is set up between two players, the rules of which are as follows:

1. Each player chooses a point on the triangle, including the sides. The winner is the one who obtains the highest value by adding the distances from the chosen point to the sides of the triangle.
2. In case of a tie, the winner is the player who has the greatest distance from the point to the side of the triangle.
3. In case of a new tie, the winner is the one with the greatest distance from the point to the line through the centre.

How should the point be chosen to win?

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Solution

We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.