Hacker News Reader

Comment by Kiro | Hacker News Reader

If the board is infinite, can't the angel just jump in one direction forever?

No. Lets say we're playing with a 2-angel. Lets place an angel, and without loss of generality assume the angel is moving left:

    _ _ _ _ _ A _ _ _ _ _

Then, perhaps the Devil goes here:

    _ D _ _ _ A _ _ _ _ _

Angel moves:

    _ D _ A _ _ _ _ _ _ _

Devil places:

    _ D D A _ _ _ _ _ _ _

The angel must now change direction. For any given direction and any given angel power, as long as the devil starts placing pieces far enough away he can force the angel to change direction.

Couldn't the angel calculate when it's necessary to change direction and then do so, forcing the devil to begin constructing a new trap, and then keep repeating the same behavior? I'm sure there's a reason why this obvious strategy wouldn't work, but I don't quite see it.

It has been proven that a 2-Angel can.

The difficult part of the problem is proving your argument to others. There-in lies the difficulty of mathematical proofs.

I only read the part about pretending the left half is blocked and using the left wall as a guide, which seems a lot more specific than just "go in one direction until you're approaching a trap and then change." I understand that proofs are much harder than intuition, but it seems that the angel has such an advantage of choice that it would be easy to prove. At any point, the 2-angel can move to 8 spots on an infinite plane and the devil can block 1. I wonder how constrained the problem could get for the angel to have a winning strategy. Let's say the angel could only move in two directions. It would seem intuitively that this would still leave enough room to avoid traps. Devil starts creating a trap in the up direction, angel moves right. Devil starts constructing a trap in the right direction, angel moves up. If it were possible to constuct a trap that blocked both directions with an unknown rate of movement on an infinite plane, it might lead to some interesting applications to other things, but my intuition says it's not.

I think if the Angel was restricted to move top and right, it could not escape.

Conceptually, the devil will start constructing both traps at the same time at half speed, and decide which one to put more work in as the angel approaches.

> Let's say the angel could only move in two directions. It would seem intuitively that this would still leave enough room to avoid traps.

Nope. Even if you restrict the angel to 3 directions (up, left, right), the devil has a winning strategy. This is mentioned in the linked article:

> If the angel never decreases its y coordinate, then the devil has a winning strategy (Conway, 1982).

There's a simple, informal proof in the references:

Conway, H. "The Angel Problem" http://library.msri.org/books/Book29/files/conway.pdf

Thanks for the clarification. I do wonder what other kinds of things this math could apply to. It's pretty abstract, but if you want to direct an unpredictable agent toward a certain behavior, knowing where to place control mechanisms might be interesting.

This is remarkably similar to the game of go. The thing is that the devil can place pieces anywhere. They don't have to be close to the angel. So he really only has to calculate the arc that is possible for the angel to travel in and play pieces somewhere around there. As the angel gets closer, the devil needs to fill in the spaces between his pieces. But because he doesn't have to construct a complete wall, his initial moves are very inexpensive. A single piece can be used to potentially block off a fairly wide arc. The trick is to progressively restrict the options of the angel until there are none.

If the Angel backtracks, this is good for the devil because it's amost like getting a free move. You can imagine that if the angel moves right and then moves left, he's back in the same place but the devil has place 2 pieces. So the Angel is a lot more restricted than it first appears.

not 8; 25

Sorry! 8 is for the 1-angel. With a 2-angel you can go one step beyond in each, so 16.

Not 16 either. 25. Or maybe 24 if standing still is forbidden.

I tought angel had to move like a chess king or extend the movement. That restricts the extra 9 moves.

But I see it considers movement as multiple movements of a chess king, so you're right. It doesn't have that restriction!

Every move increases the area needed to be blocked.

Starting off, if the angel can move 1 every 1/9th of a turn, a devil could block off a 3x3. Every subsequent full turn would add 2 that the devil would have to block off.

So for move rate M on an NxN grid (which can represent any wandering linear path) necessary to trap an angel is 8+2NM (interestingly, close to 13). If an angel moves at all, the devil cannot trap it without player error.

But what if I put a block where you are about to jump?

The angel is not allowed to jump onto a block; it could not decide to jump where you placed the block.

Reply on news.ycombinator.com