Bellman's lost-in-a-forest problem
What is the best path for a lost hiker to follow to escape from a forest of known shape?
Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman. The problem is often stated as follows: "A hiker is lost in a forest whose shape and dimensions are precisely known to him. What is the best path for him to follow to escape from the forest?" It is usually assumed that the hiker does not know the starting point or direction he is facing. The best path is taken to be the one that minimizes the worst-case distance to travel before reaching the edge of the forest. Other variations of the problem have been studied.
Although non-contrived real-world applications are not apparent, the problem falls into a class of geometric optimization problems, including search strategies that are of practical importance. A bigger motivation for study has been the connection to Moser's worm problem. It was included in a list of 12 problems described by the mathematician Scott W. Williams as "million buck problems" because he believed that the techniques involved in their resolution will be worth at least a million dollars to mathematics.
Known cases
Although it is not known how to find an optimal solution for an arbitrary shape, the optimal solution is known for some special shapes and special classes of shapes:
If the forest contains a 60° rhombus whose long diagonal is a diameter of the forest, then the optimal escape path length is the diameter, and walking in a straight line for this distance will provide an optimal escape. This case includes, for instance, a circular forest.
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