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Home Mathematics Miscellany The Bellman Forest Problem
The Bellman Forest Problem

My Master's research focused on a curious problem, easy to state, but challenging to solve.  I've included links for the paper and its web presentation.

The problem was posed by Richard Bellman in 1956: You're lost in a forest of known dimensions but no way of telling where you are or which direction you are facing.  What is the shortest path guaranteed to get you out of the forest?

Stated another way, what's the shortest path you can draw which can't be placed on the map without touching the border?

Interesting features of the problem:

  • We have only three solution paths so far - a straight path, a wishbone shape due to Zalgaller, and a zigzag due to Besicovitch (only recently proved).
  • For each of these solutions we know a class of forests for which they are optimal.  But we haven't determined how many other types of forest might also be solved by these paths.
  • For one rectangular forest, two of the paths are optimal.
  • There are some forests for which we can show that none of the three known solutions is optimal, but we don't know what the best path would be.

Sound like fun?