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[Eagle of Fire]

Angel cannot win unless you can define infinity.

Angel lose.

End of story.

[arete]

Mmm, chessboard of infinite size? If length of side is n, then n is infinity, yes? But, the number of squares on the board is also infinite, as is the surface area of the board, n x n. What you're essentially saying is there would only be one square left, for the piece A to stand on: Therefore d must be able to remove n-1 squares from the board, in order to win. This is a paradox, because if that's possible, infinity - n - must be quantifiable. Because otherwise there will always be more squares for A to move to. Infinity minus one is still infinity. If you can count them, however, it's not infinity. The board is huge, of course, but not infinite. It can be argued that the universe itself is not infinite, so now we're talking quantum? In which case, I'd rather talk about something else.

[The Fifth Horseman]

If the devil focuses on building a large "moat" of pits and then takes out the borders of the large "island" formed that way before the angel can run out... yeah, he can win.

[arete]

Was I wrong in understanding that angel can jump over any number of empty spaces?

[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.

[Juanca]

Who is the first player? who would like to play against Devil?
If it is God then the possibilites are infinite :amused:. But if there is no first player then Angel cannot move whatever is the N NUMBER.

[AlumiuN]

The angel will be able to keep moving in any given vector as long as it can move more than one square per turn. Because the devil can only remove one, the angel can simply jump over it if it's [N] is higher than the row of holes, otherwise it can simply move off in another direction. And even if the angel only has an [N] power of 1, it would take a long time and some exceedingly good forward thinking by the devil to trap it.

[Japo]

A fine point is that, if the angel needs to keep on going infinitely to win, he won't ever win. But this belongs to the infinity discussion, and maybe heavenly beings play with different rules. L0L

Originally Posted by The Fifth Horseman

If the devil focuses on building a large "moat" of pits and then takes out the borders of the large "island" formed that way before the angel can run out... yeah, he can win.

Can he? The number of tiles he 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.

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.

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.

I'm not going to think of all the possible strategies now, so I don't have an answer, but I wouldn't bet for the devil--perhaps out of principle. :bleh:

[dosraider]

The angel can't win.
Period.
The devil will get the angel and the devil will screw the angel.

Devils cheat, ya know, that's why they are devils.
:nuts:

[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.