Jordan’s theorem is famous for being particularly simple to understand, but particularly difficult to demonstrate rigorously. Take a sheet of paper and a pencil, then draw a continuous line (i.e. without lifting the pencil) which starts from a point and returns to this same point without ever crossing itself. In short, a line that loops. The theorem then states that the line divides the surface of the paper into two zones: the inner zone and the outer zone. Although this result seems like common sense, it was not fully demonstrated until 1887 by mathematician Camille Jordan.

This result is also sometimes called the “shepherd’s theorem”, because to guard their sheep, shepherds have every interest in using Jordanian fences.

The image below shows such a fence seen from the sky. We can be sure that the shepherd has done his job well: the fence forms a loop and therefore demarcates a single interior area from which it is impossible to exit without jumping over. Unfortunately, clouds hide certain areas and the shepherd no longer remembers the shape of his enclosure very well.

For each of the sheep A to F, is it possible to tell the shepherd whether it is inside or outside the enclosure, despite the presence of clouds?

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