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Feedback for Foxes

rabbit and fox, illustration

Credit: Getty Images

The early history of differential games gave us the parable of the homicidal chauffeur. The chauffeur, it seems, wants to use his car to collide with a person running on an infinite plane. The car is faster but less maneuverable than the runner. The general question is: What is a good strategy for each?

In this Upstart Puzzle, I will simplify and discretize the problem in order to discuss the interrelated questions of feedback, impossibility, and time complexity. To begin, consider the scenario of the figure in this column. There is a scared rabbit inside a straight corridor C that is glass-lined so the position of the rabbit is known to any observer. A fox starts outside the corridor at a point such that the line segment of length D from the fox to the rabbit is perpendicular to C (hereafter, the perpendicularity condition). The fox's goal is to be able to catch the rabbit, which it can do from at most one unit away.


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