Angel Problem(en.wikipedia.org)
en.wikipedia.org
Angel Problem
https://en.wikipedia.org/wiki/Angel_problem
35 comments
Managed to find a strategy pretty quickly: begin from the outside, opposite of the direction you want the cat to go in; then fill until you have 1 exit in that direction and begin to fill in the exits while the cat travels to the one open route, and when it's one step away you close it. You then have many more moves before the cat can get back to open area, in which case you can repeat until you have a closed loop.
I eventually found a strategy of basically trying to fill every "even" space around the edge, and then only filling in the odd ones when the cat was 1 space away from entering the edge-ring. This would allow me enough time to enclose the entire board, after which trapping the cat into a single space is just a matter of time
The Angel (cat) in this example is incredibly naive which made it easy to realize that strategy would work. Would definitely make for an interesting 2 player game!
The problem, with 1-angel, feels very go-like.
I think it's worth noting here that the hex grid makes the devil's job significantly easier, since a square grid space borders 8 other squares for a 1-angel, and 24 squares (or 16 outer squares) for a 2-angel, where the equivalent hex grid figures are 6 and 18 (or 12).
For a strategy based on constructing a perimeter and filling in its gaps as the angel approaches, it should be obvious why a smaller perimeter for the angel is an advantage.
Also, it's pretty unsatisfying playing against that cat, since I can't shake the feeling that it's making some really dumb moves. I'd feel like I was actually learning something if I thought the cat knew what it was doing. :/
For a strategy based on constructing a perimeter and filling in its gaps as the angel approaches, it should be obvious why a smaller perimeter for the angel is an advantage.
Also, it's pretty unsatisfying playing against that cat, since I can't shake the feeling that it's making some really dumb moves. I'd feel like I was actually learning something if I thought the cat knew what it was doing. :/
Has a nice relation to the board game Hex: https://en.wikipedia.org/wiki/Hex_(board_game)
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Mathé's 2-Angel proof is really nice, or at least the summary is appealing -- he imagines a 'nice' devil, shows it can be beaten, then proves that if you can beat the nice devil, you can beat the mean one.
This is one of my favorite problem solving strategies -- reducing to a more obvious solvable situation, then filling in the chinks and gaps to expand back.
This is one of my favorite problem solving strategies -- reducing to a more obvious solvable situation, then filling in the chinks and gaps to expand back.
The Kloster solution[0] solves it in a more direct way, by showing that for each action the devil takes the angel can take a counter action by limiting it's own moves.
In the one proof is the realization that sometimes it's easier to solve a smaller problem then prove equivalence to a harder problem, and in the other is the realization that sometimes voluntarily reducing the number of actions can lead to a simpler solution. Both are pretty good tools to have under one's belt.
[0] - http://home.broadpark.no/~oddvark/angel/kloster.html
In the one proof is the realization that sometimes it's easier to solve a smaller problem then prove equivalence to a harder problem, and in the other is the realization that sometimes voluntarily reducing the number of actions can lead to a simpler solution. Both are pretty good tools to have under one's belt.
[0] - http://home.broadpark.no/~oddvark/angel/kloster.html
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 _ _ _ _ _
_ D _ _ _ A _ _ _ _ _
_ D _ A _ _ _ _ _ _ _
_ D D A _ _ _ _ _ _ _
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.
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.
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.
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.
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.
> 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
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.
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.
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.
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.
not 8; 25
Sorry! 8 is for the 1-angel. With a 2-angel you can go one step beyond in each, so 16.
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From the Wikipedia description it is bit vague how many blocks the devil put down each turn. Is it just 1? The same as angels power?
> The devil, on its turn, may add a block on any single square not containing the angel.
That phrase can mean both 1 block and one block only, or it can mean one block per square, on as many squares as needed as long as the block is not placed on a square containing an angel or another block.
Particularly when read without emphasis.
I think I leaned towards an ambiguous reading because I couldn't understand how one block per turn was enough to ever win.
Particularly when read without emphasis.
I think I leaned towards an ambiguous reading because I couldn't understand how one block per turn was enough to ever win.
Just one.
Amusingly, a variant of this is present in Beyond Zork toward the end of the game.
Surprisingly tricky, even with a 1-angel!