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[DoomYoshi]

The devil will always win. If he starts building a moat 1000 squares thick, and he builds it so far away(approx. 100 000 trillion squares away) then the angel won't be able to escape the moat before it is completed. Then the devil just closes in on the devil and voila! Now you see that evil will always triumph because good is dumb
Except the moat will get huger as the distance away you build it gets huger, the horseshoe shape will have to be used... maybe this isn't as simple as I thought...

[_r.u.s.s.]

yes.. and in case of horseshoe, the angel can also move backwards, or in spiral..

[Japo]

Originally Posted by DoomYoshi

The devil will always win. If he starts building a moat 1000 squares thick, and he builds it so far away(approx. 100 000 trillion squares away) then the angel won't be able to escape the moat before it is completed.

Er... The only thing that's been proven in this topic is that what you say is wrong:

Originally Posted by Japofran

The number of tiles he [the devil] must remove to create a sqare island of side D surrounded by a moat N tiles wide, is 4DN+4N². That takes him an equal number of rounds. In that time the angel can traverse 4DN²+4N³ tiles.

Even if N=1, the angel could traverse 4D+4 tiles by the time the devil builds a moat 1 tile across surrounding a square D tiles across.

No matter how big you make D, it will only take the devil longer to build, and give the angel more margin.

[_r.u.s.s.]

but as devil builds, angel has moves as well, so he doesn't have to build all the moat at once, you know, if angel moves forward and devil builds moat- in front of angel, then angel sees that the moat is being built and can change direction, but then devil can start building a moat in different place too.. it's more about finding right algorythm or random sequence for the angel

[Japo]

Originally Posted by _r.u.s.s.

but as devil builds, angel has moves as well, so he doesn't have to build all the moat at once, you know, if angel moves forward and devil builds moat- in front of angel, then angel sees that the moat is being built and can change direction, but then devil can start building a moat in different place too.. it's more about finding right algorythm or random sequence for the angel

Of course the devil would win if he plays an optimal strategy and the angel plays dumb. And vice versa. But you want to prove that the angel or the devil can win no matter what the other side does.

I said there are no N small enough and D (trap size) big enough that the devil could build a closed moat around an area larger than the total angel's mobility in that time; but I didn't say this means the devil can't win. (Although maybe it does but I haven't thought of a demonstration.)

[_r.u.s.s.]

no, i think that devil maybe is able to win even thought that angel choses an optimal strategy

[Japo]

Prove it and you'll earn $1,000.

[AlumiuN]

Well, the devil can win if the angel makes some very stupid moves. Like moves into a space where the devil can instantly trap the angel. If the angel is playing appropriately, then the devil cannot win.

[Japo]

Prove it and you'll earn $100.

[blackboard]

Originally Posted by The Fifth Horseman

The angel's [N] can be anything from 1 to infinity. For angels with low [N], it's definitely possible to trap them.

Are you sure? I thought about it a bit, and I tend to believe that the devil cannot capture the angel even if N=1 (assmuning the angel has total knowledge about what is happening on the board).

This can be seen by counting: To capture the angel the devil has to surround it some closed shape. We assume for simplicity it is a squared rectangle. Of course a circle would be better, but it is more hard to calculate with, and a square is less than a factor of two worse.
Now when the devil builds a squared rectangle of length l, it needs N*4*(l-1) moves (and a few more if N>1, too lazy to derive the exact formula). The minimum length is obviously 3, the angel has to fit inside.

Unfortunately (for the devil) this gives the angel more than enough time to leave the rectangle. And as we can safely assume that he will need more than N*2*(l-1) moves for any kind of closed shape of maximum diameter l, the angel can escape any shape the devil might come up with, simply because it will be unfinished when the angel reaches the border. Without enclosing shape the angel can move infinitively, of course, as the chessboard is infinite.

This assumes the angel knows the moves of the devil and plays perfectly (as fitting for an angel), but then the devil cannot win. Closing the shape simply takes too long. If a part of the shape is closed the angel will head in the other direction. And even building a shape partially (setting every second stone e.g.) and the filling when the angel approaches is not fast enough, if the angel races consequently towards infinity (as it must when playing perfectly) the devil cannot build fast enough.