The angel wins

TL;DR

This paper proves that for sufficiently large jump distance J, the angel can always evade the devil on an infinite chessboard, confirming a long-standing conjecture and offering a hierarchical strategy that may aid future generalizations.

Contribution

It introduces a hierarchical strategy demonstrating the angel's winning condition for large J, addressing a thirty-year-old conjecture in the angel-devil game.

Findings

For large enough J, the angel can always evade the devil.

The hierarchical approach may be applicable to other pursuit-evasion problems.

The result confirms the angel's winning strategy for sufficiently large jumps.

Abstract

The angel-devil game is played on an infinite two-dimensional ``chessboard''. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil's turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal, vertical or diagonal) to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared simultaneously, some of them prove that J=2 is sufficient (see the Wikipedia on the angel problem). Still, it is hoped that the hierarchical solution presented here may prove useful for some generalizations.