# King and Devil problem

On a unlimited two-dimensional plane, the plane is separated into two-dimensional grid point by line $x=k$, $x=-k$, $y=k$, $y=-k$ ($k$ is integer). There is a game like this : A king could move to any one of the neighbor $8$ grid points from the current grid point; The devil could destroy any grid point except the grid point where the king is located; The king couldn't move to the grid point which has already been destroyed by the devil. The initial grid point is $(0, 0)$. The king moves firstly, then the devil moves. The question is :

a) what A should satisfy so that the devil has a strategy to win to make the king constraint under line $y=A$ .

b) what B should satisfy so that the devil could constraint the king out of the region $x\geq B$, $y\geq B$.

c) Prove that integer $C$ exists so that the devil has a strategy to win to make the king constraint in the region $-C\leq X\leq C$, $-C\leq y\leq C$.

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## 1 Answer

This problem of John Conway's is discussed at this link.

EDIT: On closer inspection, I see that all it says at that link about the question under discussion here is that it was answered in 1982. It may be that the details are in the Berlekamp, Conway, Guy book, Winning Ways. There is a lot of information to be had about (variations of) this problem by searching the web for $$\rm Conway\ angel\ game$$ possibly including a complete solution of the current question, but I don't guarantee it.

MORE EDIT: The details are indeed in Winning Ways, in Chapter 19, The King and the Consumer.

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Yes, the problem was solved by four groups independently: the angel wins. See this link. – Joseph O'Rourke Jan 30 '12 at 13:47
@Joseph, if I remember right, an angel that can move 2 squares in any direction wins, but a king loses on any board larger than $32\times33$, and OP is asking about a king. – Gerry Myerson Jan 30 '12 at 23:16
My mistake, sorry! – Joseph O'Rourke Feb 1 '12 at 1:04