Geodesic
The Angel Problem on Triangular and Hexagonal Boards
Publications de l'Institut Mathématique, _N_S_118 (2025) no. 132, p. 17

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DOI

The Angel problem, introduced by Conway in 1982, is a two-player game played on an infinite board where an Angel of power $k$ competes against the Devil. We examine variations of this game on triangular and hexagonal boards. Using Máthé's proof technique originally developed for the square board, we prove that the King (Angel of power 1) can win on a triangular board. Through a mapping between hexagonal and square boards, we then establish two results for the hexagonal board: first, that the Devil can defeat the King, and second, that an Angel of power 2 can win. Our proof for the triangular board involves analyzing the perimeter bounds of connected sets and developing a wall-following strategy for the King, while our hexagonal board results utilize transformations that map to known results from the square board case.
DOI : 10.2298/PIM2532017B
Classification : 91A43, 91A05
Keywords: combinatorial game theory, angel problem, infinite games, geometric graph theory, triangular grid, hexagonal grid 
Vukašin Babić. The Angel Problem on Triangular and Hexagonal Boards. Publications de l'Institut Mathématique, _N_S_118 (2025) no. 132, p. 17 . doi: 10.2298/PIM2532017B
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