Shortest escape path for the golden gnomon
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What is the shortest rectifiable escape path for the golden gnomon, the isosceles triangle with unit equal sides and 36-degree base angles? The target compares all admissible competitors with a symmetric seven-piece candidate of length about 1.282676025459.
Why it matters and what counts as progress
Why it matters: An exact optimum below the earlier solved angle range would distinguish a global escape path from a numerical candidate.
What counts as progress: Prove a global lower bound matching the candidate or solve the corresponding sharp worm-cover problem for this triangle.
How the problem developed
- 1956 · Problem formulation [1]
Richard Bellman Formulated the minimax lost-in-a-forest problem: find a path that guarantees escape from an unknown starting point and heading.
- 2004 · Survey [2]
John H. Finch and John E. Wetzel Surveyed the lost-in-a-forest problem and its unresolved triangular cases; this predates the exact golden-gnomon solution.
- 27 Jul 2026 · Exact solution [3]
Alexander Temerev and Alessio Doria Reported an exact shortest escape path for the isosceles golden gnomon with unit equal sides and 108° apex angle: a symmetric seven-piece path of length 1.282676025459…; Lean checks the finite certificate families and discrete identities.
AI contribution
The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon
Temerev and Doria report that the minimum escape-path length for the golden gnomon, an isosceles triangle with unit equal sides and 36-degree base angles, is 1.282676025459… . A symmetric seven-piece path attains the value, and their lower bound covers every rectifiable competitor.
Sources
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