John Conway proposed a game on the infinite square grid. Rules: 0. All squares initially white. 1. angel can in 1 move fly a taxicab distance A from wherever she is, to a white square. 2. devil can in 1 move color any B squares black. 3. If the angel has no white squares within range to fly to, then angel loses. If it can escape to infinity, angel wins. Who wins? The problem with B=1 remained open for about 30 years, then 4 people [Gacs, Kloster, Mathe, Bowditch] independently solved it, showing that with any A>=2 and B=1, the angel wins (but with A=B=1 the devil wins, shown by Berlekamp). But for some reason the problem with B>1 was not addressed. http://home.broadpark.no/~oddvark/angel/index.html http://en.wikipedia.org/wiki/Angel_problem The real truth is, if B is large enough compared to A the devil wins, if A is large enough compared to B the angel wins. The real question is: which (A,B) are which? If B>2*A*(A+1) the devil wins in 1 move. If A>4*B the angel wins, as far as I can tell from generalizing Mathe's proof. Gacs' proof also shows directly that there exists a positive constant so if A>const*B then angel wins. There is a large gap between these.