The Angel Game in the Plane

The angel game, as described in [BeCG], has two players the “angel” and the “devil” who play alternately on the 2-dimensional integer lattice, Z. We refer to lattice points as vertices, and write o = (0, 0) for the origin. The angel has a certain fixed power p. (We refer to it as a p-angel .) It starts the game at the origin, and at each play moves to another vertex so that the change in each coordinate is at most p in absolute value. On its turn, the devil can “block” any vertex other than that currently occupied by the angel. Once a vertex has been blocked, it remains blocked forever. The angel is not subsequently allowed to visit any blocked vertex. The aim of the devil is to trap the angel so that its only legal move is to remain where it is. (Of course, it would be sufficient to trap it within some bounded set.) Since the game is infinite, we speak of the angel as “escaping” if it never loses, i.e. is never trapped. (There is a finite version played on a square board, where the angel wins if it reaches the edge of the board starting from the centre.) In [BeCG] it was asked if there is some p ∈ N such that the angel of power p can always escape. We shall show that, in fact, the 4-angel has a computable winning strategy. A formal statement for that case is as follows. Given n ∈ N, write I(n) = [−n, n] ∩ Z and W (n) = (I(n)). We write σ(n) for the location of the angel at time n, and ∆(n) for the set of all vertices that have been blocked at time n. Thus, σ(0) = o, σ(n) / ∈ ∆(n), and σ(n + 1) − σ(n) ∈ W (p). We start with ∆(0) = ∅, and ∆(n+ 1) is obtained from ∆(n) by adding a single vertex other than σ(n). Thus |∆(n)| = n.