The Angel Problem [Game Theory] -

The Angel Problem [Game Theory]

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A fascinating Game Theory problem proposed by John Conway.

Original Paper:
2-Angel Proof:

Bass Loop – Wooseong


  1. If the angel goes in a straight lone forever he will never stop

  2. Couldn’t the devil just go super far away and build a theoretically infinite square (however thick the angel can move) and the angel couldn’t get past it?

  3. That character he has is animated like a Minecraft story mode character

  4. If it's finite, the devil just has to keep moving until all the boxes besides one is filled

  5. Well, it's true the devil gets free moves if the angel comes back on a space it already visited, but only if the devil effectively blocks those spaces, so if the devil starts creating a wall ahead, the angel can just go back and nothing the devil has done will have an impact

  6. i was thinking what if the devil started like a centillion blocks out and made a wall fifty tiles big on all four sides and then started filling it in. wouldnt that work? or maybe with an even bigger number?

  7. With 1 move space devil wins everytime, with two or more it depends on wether the angel can jump over blocked spaces. If yes then angel wins unless the devil somehow backs them into a pre laid trap same for everything above 2

  8. before i watch any more than 0:40, heres my guess:
    the angel just continues on into one direction of the infinite grid without turning back to the tiles the devil stole behind him, easy

  9. Infinite World = Angel Wins
    Finite World = Devil Wins

  10. “The angel would never want to move to a space that she could have landed on previously. It’s impossible for that to be helpful.”

    Ok but at 2:08 you show a situation where it would CLEARLY be beneficial for the angel to move back to a previous square.

  11. So what if the devil took his time to make a 1/2 of k sized wall around the angel and then slowly fill it in

  12. "You're an angel on an infinite grid."

    "Aww, shucks!"

  13. 2:13

    According to the formula being shown, the distance for K=3 should be 3*10^32.

  14. I don't understand why you think playing a move next to where the angel was before can't be benifitial. While the devil has to "waste moves" on the inner squares, the angel can jump around until he ACTUALLY fills up the bottom half or something, and then she can just jump away. It's like giving your opponent tempo on chess. Why do you just throw this idea away?

  15. everyone is like "the devil can create a box to trap the angel in" as if the angel is even going to go towards the box

  16. Even knowing that it's true I find it very difficult to believe that the angel could ever win. Surely the devil could always just build a box with thickness k far enough out that the angel wouldn't reach it in time and once that's complete the angel is stuck in the box while the devil slowly fills the whole thing in?

  17. Code parade: you're an angel
    Everyone: *blushing intensifies*

  18. love ur work, here's a comment to boost you for the algorithm

  19. In theory, the devil could win using hypertasks.

  20. It would be cool to see this problem with slightly different conditions, such as the devil being able to block multiple squares instead of only one.
    Also I still feel like the devil can win, it would just take an extremely long time. Depending on the value of K, the devil could build an entire fenced area instead of a single wall, far enough from the angel that it would be finished by the time she reached it. Since the angel needs to survive an infinite amount of time, the only thing guaranteeing her safety is an infinite amount of space, which she has. But if you go far enough you can create the walls all around the angel that are so thick she can’t get past them. By this point the angel has technically lost, she will wander this area as the devil shrinks it more and more.

  21. But couldn't the devil start building a wall like billions or trillions blocks away and just make a full square? The Angel wouldn't be able to move anywhere but in the square, then the Devil would just fill it up and boom, Devil wins easily, yes, it would take years probably but still…

  22. "But thats just a theory. A GAME THEORY. Thanks for watching"

  23. i thought the devil can build a great wall all around the angel so she is trapped and then fill it in (of course the wall would have to be super far away and think so the angel can't go through) but i don't know. can you guys prove me wrong?

  24. Looks like its simple math: 1st Devil task is to build closed wall when Angel is inside this space. Minimum of turns takes the wall of nearly circle shape. Its length = 2*pi*R – and its also a number of turns it takes the Devil to build it. But Angel could escape this space by walking along a radius, & his speed is 10 cells per turn. So the time Angel needs to leave this circle space is R/10.
    So, because R/10 < 2*pi*R, Angel always can escape. If he sees all blocked cells and want to escape. Am I wrong?

  25. But the devil could build a massive wall around the angel, farther out than the angel could get to before the entire wall is done, then fill in the entire inside of the wall. Checkmate

    Edit: wait. It couldn't build the wall fast enough. So unless the angel is dumb, it can't be captured

    Edit2: I have an idea. It's complicated

  26. I would like to see a logical proof that the angel does not benefit at all when going to a spot the angel could have gone before.

  27. This is really dumb. The game theory falls apart when you remember that the angle can, in fact, move over those spaces behind her.

    The whole "the angel wouldn't move anywhere she could have already moved to" bit makes no sense. If the path in front is now blocked, she can just turn around or move to the side. Those "blacked out spaces" aren't actually blacked out. She can use them still. You're just saying they are options anymore.

  28. The mundane rose covalently confuse because gun principally visit out a cool cupcake. imported, abnormal sword

  29. The “nothing helps the angel” argument is misleading. Why can’t the angel just turn around?

  30. "When you don't have problems. Make up problems."


  31. I'm no mathematician, but couldn't the devil start a border far enough away that he would have time to completely enclose the angel (with 'walls' wide enough that the angel can't pass it with it's limited movement) and then it's essentially checkmate, start filling in the area until no space remains.

    Edit: I think I figured out why that wouldn't work. The stipulation was that if k=2 or more the angel can win. I'm guessing that the wider the devil has to build his wall the amount of turns it would take to complete grows at an exponential rate that cannot overcome the speed of k=2

  32. Even if you say it would only help the devil, when running into a wall, moving back or to the side until you break it could also work. While the devil is trying to stop you from going up and sideways. You could continuously move down and to the other side every now and then to break it

  33. Can't the devil build a huge circle of removed tiles where the angel can't reach (in time) and then just wait?

  34. I wish he would do a 3d model, but everyone says god sees eternal wouldn't that mean he is a fourth dimension creature because the fourth dimension has lots of evidence to be time so if the devils minions ever got a body and became good and got higher rankings and learned how to time travel and then instantly out of nowhere betrays good again couldn't that being attack the good with all its power and destroy one angel? but if god saw eternally from the beginning he knows what will happen and needs to happen. please let me know your thoughts, I typed sloppy and some bits may take some time to think about but the idea was fresh didn't want to lose it

  35. If the Devil play's it smart, there is no way out.
    There is infinite space. The Angle can only cross 10 squares.
    The Devil just has to make a huge circle or square around the Angle that is a minimum of 10 blocks wide. Of course it has to be massive so the Angle doesn't reach it before i's done, but as there is unlimited space and time, that is not a problem. Once the huge square or circle is complete, the Devil can start filling it in one by one. Soon enough, the Angle is trapped.

    So the answer to the question is yes, the Angle can escape, but only if the Devil is stupid enough to let it happen.

  36. Imagin the guy sitting in his room just going “mhhhaha”

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