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.
Approaches
A proven solution is only known for a few shapes or classes of shape, such as regular polygons and a circle. In particular, all shapes which can enclose a 60° rhombus with longer diagonal equal to the diameter have a solution of a straight line. The equilateral triangle is the only regular polygon which does not have this property, and has a solution consisting of a zig-zag line with three segments of equal length. The solution for many other shapes remains unknown. A general solution would be in the form of a geometric algorithm which takes the shape of the forest as input and returns the optimal escape path as the output.
References
- Bellman, R. (1956). "Minimization problem". Research problems. Bulletin of the American Mathematical Society. 62 (3): 270. doi:10.1090/S0002-9904-1956-10021-9.
- Finch, S. R.; Wetzel, J. E. (2004). "Lost in a forest" (PDF). American Mathematical Monthly. 11 (8): 645–654. doi:10.2307/4145038. JSTOR 4145038. MR 2091541.
- Williams, S. W. (2000). "Million buck problems" (PDF). National Association of Mathematicians Newsletter. 31 (2): 1–3.
- Ward, John W. (2008). "Exploring the Bellman Forest Problem" (PDF). Retrieved 2020-12-14.