Bellman's lost-in-a-forest problem is an unsolved minimization problem in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, 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 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 Moser's worm problem (also known as mother worm's blanket problem) is an unsolved problem in geometry formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem asks for the region of smallest area that can accommodate every p ...

. It was included in a list of 12 problems described by the mathematician

Scott W. Williams Scott Williams (born April 22, 1943, in Staten Island, New York) is a professor of mathematics at the University at Buffalo, SUNY. He was recognized by Mathematically Gifted & Black as a Black History Month 2017 Honoree. Education Raised in Ba ...

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.{{cite web , last = Ward , first = John W. , title = Exploring the Bellman Forest Problem , url=http://wardsattic.com/math/BellmanForestProblem/BellmanForestProblem.pdf , date=2008 , access-date=2020-12-14 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

Metric geometry Discrete geometry Unsolved problems in geometry Recreational mathematics