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Your link to Conway's article about the problem needs updating now that the MSRI has become SLMath. (The pathway is exactly the same, just replace "msri" in the URL with "slmath")

7:23 since when does 2x3 = 1. and 2x4 =3 ????

I figured it out. The devil wins everytime by an unforeseen flaw in the games design. Mathematically speaking, with finite moves the angel can never lose with perfect play, but the devil can place blocks _Anywhere_ on an *Infinite* grid. And this is the angels weakness. The devil simply places a block infinitely far away. And does so over and over again every move. Not even building a wall. these blocks dont technically exist within the Angel's span of interaction since the angel can NEVER reach them without an infinite number of moves. Because no matter how many positive numbers you add in distance, they can never be equal to infinity unless multiplied by infinity which is why the angel cannot move towards this wall. With finite moves, nothing effectively happens, the angel can make any amount of moves less than infinity and nothing happens. But if the game goes on for an infinite number of turns, the angel loses. Because an infinite number of turns is equivalent to an infinite number of spaces. And the devil removes one square of available space each turn. Meaning an infinite number of the infinite squares are now blocked. Preventing the angel from moving. We cant effectively observe a "when" for the Angel's loss happening, since within our reality infinity is inobservable, but this sets a precedent for the game that transfers to ANY board size. The game has infinite moves for each side, we can never observe that limit be reached, because it's not a limit, it's an absence of limit, but this means no matter how big or small the board, with infinite moves the devil wins. How it wins can never be observed or calculated since we have no effective measurement to infinity, but its comparable to ANY finite board we can measure. Given infinite moves, which each side has, the devil allways wins on a finite board, because every square can be blocked. A million squares is less than infinite squares, slowly the board fills up. Until the angel loses. This is where its tricky, because infinite time is also inobservable, so we cant "see" a game on an infinite board be played, we can only see a game on a finite board be played. We can make that board bigger as much as we like, but it will never be infinite. Eventually we hit a conceptual limit of reality, be the board made of matter or code, we can only make it so big in this existence, and any number of spaces we can make it is less than or equal to the turns we can allot it. Thus, since the game _allows_ infinite turns, no matter the board size, infinity or finite, the devil wins. We cant observe _how_ the devil wins on an infinite board, but it does.

Couldn't the devil just go even further out & make a thicker wall

The angel with power of 2 would win because the fact that the devil can make a wall in a 4th of the moves in one direction that means he cant make 1 wall in 4 directions in less than the equal amount of moves I originally thought the devil would win but it makes sense when you think that the devil has to cover 4 different sides when the angel only needs one

i dont understand. literally any of this augshjhshdd

How does the nice devil win. Wouldn't the normal angel just be able to move in a circle since not all of its squares are blocked?

The real question is does an Angel that alternates between 1-2 power ever get caught

Conway made this?

Doesn’t the runner have the same ability as the angel of power 1!? Also how do we know angel of power 1 loses, we only know that the fool of power 1 loses? Also if the nice devil viability = devil viability, how would the nice devil trap the angel of power 1?

I found this problem and proof very interesting. I found your explanation of the proof Fully frustratingly terrible. You did not bring me on the journey of understanding how and why the optimal strategy is optimal. you only communicated to me some of what the optimal strategy was. and as this is not a real game that real people play, that information is more than useless. and not what I want from a video like this. Its taken me 20 minutes to get to this level of articulation of how this video made me feel. I guess i've been fortunate to never have had a bad teacher for something I care about, as I imagine that is what I'm feeling now.

Follow up question, how many devils are needed to win against one angel of power n? As noted by another commenter, a devil (or group of devils) that can burn an n by n area in one turn can use the same strategy against a power n angel as one devil uses against a power 1 angel, just with a scaled up grid, so the upper bound is n^2, but can we do better?

I also love listening to something I don't understand while eating.

If the board is infinite and the devil can move infinitely far, couldn’t I have just built a huge box that the angel couldn’t reach until it is thick enough to stop it from getting through ?

This is not chess. You have lied.

I just realized something that makes this problem a bit melancholy: The Devil's work is done when it wins. It can stop once it's trapped the Angel. But the Angel's work is only done when it loses. If the Devil can't win, the Angel has to keep running forever.

So this makes sense exept for the statement that if the nice devil loses the normal devil loses. For instance the runner will lose to the normal devil no matter what by the devil creating a box to trap them. This same thing will occur when an angel can move however many squares the devil needs to go out an insane distance and create a barrier of 2 thickness or whatever the jump distance of the angel is and he will win every time

Just realised the final examples are actually pretty simple. Nice devil is just to prevent trapping the runner in burnt box so the only way for the runner to lose is to loop back to the start. Assuming the left wall is already there, you would need to build the right wall + 2 squares at the top, n+2. Runner has to go the left and right wall + 2 walls to escape, 2n+2. The devil wins in n+2 turns The runner wins in (2n+2)/2 turns = n+1 turns So even if the devil moves first, the runner always wins first.

It seems like if the runner loops around an island, she loses for reasons that have nothing to do with the original problem. Also, if she offers the Nice Devil half the squares, the Nice Devil could just start building a trap 1,000 squares out, and the regular Devil could just close the loop. In that scenario, the Nice Devil's inability to trap the runner has nothing to do with the nature of the problem, and everything to do with the weird, self-imposed "no burning squares the runner has been on" rule.

Please, tone down the low tones. It's actually painful to listen, and the fix is simple

You lost me at the painting edges bit

i cannot be the only one… can someone tell me why at 10:00, there are at least N edges facing north and at least two facing west? shouldn’t it be N facing east and at least 2 facing North?

No entendi alch

6:59 this is where the fun begins

I am very confused. In your example a non-nice devil could win, easily. Consider the left column at 11:40. The left bottom of the large obstruction ends in 3 squares in a dead end. This would result in the 2-runner with the mentioned strategy stopping at either the mid or top square. A non-nice devil could simply paint in the bottom square on this turn, trapping the runner. Does the runner have to play differently than described for a non-nice devil? If so, how? Or is this solely an existence proof?

I honestly thought it said "Bagel vs the devil"

when the paper is written by someone named mathe

Turns out a fat idiot CAN beat Lance Armstrong in a bike race. You just have to tie several anvils to Lance's bike. It's still fair though, because if Lance wins, the anvils win too.

This is my attempt at the problem. The devil can only affect one square at a time, if the Angel's power is greater than one, then it can travel to more spaces than the devil can affect. Suppose the Angel travels at some angle relative to it's last position, at the maximum distance it can travel every time. In other words, the angel goes straight right N spaces, then the best approximation of 15 degrees at N spaces, then 30, over and over, essentially going in a spiral pattern, moving continuously outwards with gradual turns. If the devil makes a "solid" border (no gaps between burnt spaces), at any distance, from the Angel's path, it will not be able to construct it fast enough to catch the Angel. Why? Because no matter where the Devil start's this border, the Angel can avoid it. If we assume the Angel *has* to move in this spiral pattern with no "mixups", then there may be a way to prove that the Devil could calculate where it would need to begin constructing a border to catch the Angel's path right as it is finished (whether or not this is true is irrelevant, even assuming that it is, the Angel will still win, because:), however the Angel can do whatever it wants, it does not have to move in this spiral pattern, the minute it sees that the Devil is creating a border that would otherwise succeed in later blocking its path, it can simply swap directions, then repeat the original pattern. If the Devil begins creating a border for this new path, it can change directions again. The devil can never finish a "solid" border, no matter what it does, if the Angel follows this pattern, the devil is faced with an impossible choice. The devil must begin constructing a border, but to choose to start a border anywhere on the Angel's current path would simply prompt the angel to switch paths. Upon switching paths, it must begin construction of a new border, at which point the Angel can switch directions again. As long as the Angel constantly moves the max distance it can and adjusts its angle by a constant amount (aka, follows an outwardly expanding spiral pattern), it will always gain distance from previous "attempts" to construct borders, and will therefore never circle back into them. So, the devil cannot make a solid wall to prevent the Angel from progressing on its infinite journey. But what about a wall with gaps? moving some magical number of spaces to create a dotted line in the Angels path, then closing these gaps as it approaches some part of this large dotted line. Let's say the devil is able to do this, because if it can't, then the Angel wins, so we just need to prove that it can't. Let's say the Angel reaches this dotted wall, and it sees all of the gaps in front of it filled in, what does it do now? Well, it can't *all* be filled in, because the devil is incapable of constructing a solid border faster than the Angel could pass it. So, there must be a gap somewhere, even if currently out of reach for the Angel. All the angel has to do is follow the solid part of the curve to the closest gap, and travel the fastest route to that gap. Either: A gap does not exist, and the border is solid, but not large enough to be contain the Angel (because such a border could not have been constructed in the first place), and the Angel just needs to travel down the border as fast as it can until the Devil is unable to finish the border faster than the Angel can reach its edge, then simply walk around that edge (this is always possible even if the devil burns the exact square in front of the Angel every time it tries to approach the edge, because the Angel can simply follow that new curve infinitely, until the Devil is forced to either lose, or burn a square in the new direction the Angel would be forced to travel, thereby wasting a "turn" and allowing the Angel to pass anyway.) Or, the border does contain a gap, and the Angel can simply reach it and then walk through it. (If the devil closes it, the Angel repeats the process until it inevitably passes the wall.) No matter what kind of wall the devil tries to make, the Angel has a way to get past it, and if it has a way to get past every wall the devil tries to make, it can never be boxed in, because it would require the Angel to be unable to pass a wall at some point, which can never happen.

Intuitively, I believe that an angel of Power 1 can win by being a Fool in one direction, and then another, and then in the first one again. By zig-zagging, they become that much more difficult to actually box in, and this can be more powerful if they start in another direction before actually reaching the fence. I don't have proof, however.

When you say 'If the Nice devil loses against the angel, so does the Devil', I'm not sure what you mean by 'win' in this case. You stated that the Nice Devil wouldn't burn a square that the angel previously had access to. Ok, so if the angel decides to move back and forth between two squares that they had access to from the beginning, then they will always have a valid move no? Wouldn't this mean the angel wins? Or is the meaning of 'losing' that the angel can no longer access infinity?

The video is either incomplete, or wrong. Either there is a proof that the ND and D are equivalent in power in all instance and it simply was not included or There is no such proof In the first case you failed in not including an essential step in your proof in the second you failed by relying on a false assertion in your proof Either was 0 points

I was following along nicely right up until you introduced a new piece out of nowhere that made no sense...

- So how can an angel win with a devil? - Well if we take a nice devil and a runner-

Just call doom slayer, angel ez win

TL;DR: as I understand it. As long as Angel and Devil behave optimally and Angel is power two or more, Devil will never be able to make enough burnt squares to trap Angel in.

Fullly captured my attention while making me think... Me: "Why am i watching this?

interesting video, but to be completely honest your way of explaining the paper is confusing and hard to follow. I didn't get it until I went to the paper.

I still dont like theXiom of choice

As a layman, I see the answer as follows. The devil can only block one square per turn. The angel of power >=2 can access more than that one removed square per turn, and therefore will always be able to out-maneuver the devil.

I'm Christian now. 😇

Math is basically the enemy of visual learners. “The nice devil can’t win because this equation that means nothing to you says so.”

I don’t understand any of the math but… I do like the idea.

Is it possible to calculate the chaotic ordering of the rationals? That is, is there a finite algorithm to calculate the ordering of the first N rationals that is certainly included in the infinite branch? Also, the axiom of choice is stupid.

Amazing🎉

My favorite part of this video is the comments section filled with mathematical geniuses and intellectual giants who apparently dwarf András Máthé but for some reason or other wont publish their findings. Instead they grace us mere mortals by posting their galaxy-brain ideas in the comments section.

I don’t see how the runner always wins. The devil would just need to start arbitrarily high on the wall and build the loop in the time it takes the angel to get there. I do see how the nice devil couldn’t, because it needs to build twice as many squares due to being forced to reach the bottom of the start, but the normal devil would just close in on the runner once it is adequately high in the path.

I think the next question is "for an angel of a given power, how many squares does the devil need to be able to burn each turn in order to keep up" Also at some point in this video I started thinking of it as the coyote-and-roadrunner and it's going to be a struggle to say angel and devil again now

In the runner v nice devil scenario (or how it was described here the nice devil would actually be better off not taking the runners offer to burn half the grid for free. Just don't do anything and the runner has nowhere to run.

if we were to assume an infinite grid with infinite turns, can't the devil just calculate a huge ass outer square to create in order to block the angel's path and build that in the background? and then, the devil just needs to slowly close the square until the angel is trapped. the square would be calculated in order to: be thick enough for the angel to not cross it with its power, and far enough that the devil has enough time to build it so that, even if the angel were to walk in a straight line in a specific direction, it would not be able to make it before the devil is done. meaning, the devil would win every time, no matter the type of angel.