我們要玩遊戲嗎？ 你是一個無限網格的天使。每轉一圈，你都可以跳到K空間。 假設這個例子中K為10，這意味著你可以跳到這個邊界內的任何地方。 但是你的主要魔鬼The Devil正在觀看。 在魔鬼的轉彎處，他可以移除無限網格上任何位置的一個方格。 天使再也無法降落在那裡了。 所以問題是：魔鬼最終能否陷入天使之中，或天使是否有策略避免永遠被捕？ 這是一個引人入勝的博弈論，由John Conway本人提出的問題。 讓它如此困難的原因在於它在無限的時間和空間中發揮作用。 起初，你的直覺可能告訴你：天使當然可以贏！ 我的意思是，她很機動。魔鬼一次只能阻擋一個方格？ 魔鬼怎麼能有足夠的時間來捕捉天使？但今天，我將成為魔鬼的擁護者。 Muhahaha！ 讓我們從博弈論的角度來看一下天使的動作。 有一點需要注意的是，天使永遠不會想要搬到她以前可以降落的空間。 但證據很簡單。 如果她花了一個多的轉彎才能到達她之前可以到達的地方，那麼結果將完全相同，但魔鬼之間會有額外的動作。 所以這個策略不可能永遠幫助天使。它嚴格受益於魔鬼。 這意味著所有這些空間都有效地消失了。 事實上，它可以像魔鬼自己阻擋所有這些方格一
樣對待。 所以儘管The Devil每回合僅阻擋一個方格，但The Angel阻擋了100個。 但那沒問題。我們可以繼續走開，對吧？ 讓我們假設我們將運動限制在一定量的北方。 嗯，魔鬼對我們有一個計劃。 我將切換到K = 1以使視覺效果更容易。 但是很遠的地方，魔鬼可以開始在我們的運動錐體上方建造一堵牆，留下很大的空隙。 一旦天使到達那裡，問題會在他填滿下一個方塊時遞歸。 魔鬼有一半的時間，但也有一半的距離，所以他可以做到這一點。 這種情況不斷重複，直到牆壁完全密集，天使再也無法向北移動。這適用於任何K. 但是建造牆壁所需的距離會以指數方式增加。 好的。也許說我們不得不向北移動太嚴格了。我的意思是，我們畢竟有更多的機動性，對吧？ 好吧，康威還表明，一個嚴格增加她與原點距離的天使也可以使用一個非常相似的論點來捕獲。 魔鬼可以假裝天使的K大四倍，並使用交替的轉彎在四個方面建造牆壁。 所以我說服你了嗎？天使有什麼希望嗎？你願意把自己的靈魂賣給魔鬼嗎？ Muhahahaha！ 嗯，這是數學方面的一個開放性問題，最近才在2006年得到證實。 事實證明，天使可以獲勝！而且，令人驚訝的是。 K低至2
。 證明非常複雜，但它使用某種迷宮解決論證。 我想你可以說魔鬼在細節中。 無論如何，我只把這個問題視為一個純粹的數學問題。 所以我認為將它變成真正的遊戲真的很酷，你真的可以玩AI。 但不幸的是由於涉及的距離太遠。 這將需要一個永恆的遊戲，只是超級無聊。 儘管如此，我希望你發現這個主題和我一樣有趣。 謝謝觀看！
Comments
Of course the angel can win. She can win by not moving, thus not allowing the devil to move either. The Sans strategy.
“You’d think the angel can win easily. In fact, she can.”
First 3 minutes: the devil will always win! Last 30 seconds: the angel can win, with a k as low as 2! *does not elaborate further*
-States the angel can win -Refuses to elaborate -Leaves
Devil: I’ll build a wall up north where the angel is headed, they won’t want to move back so eventually I’ll win! Angel: simply moves to the left Devil: FUCKING- AAAA-
I love the mental image of the devil slaving away at building an impenetrable barrier with the intent to trap the angel, only for the angel to casually walk past it mid-construction. Sounds like something straight outta Looney Tunes
Plot twist: He only did this video so as he could have an excuse to say those Devil puns
I vaguely remember playing a game like this once, except it was called "Farmer and the Hog" or something like that. But instead of being the angel, you were the devil (or farmer in this case) trying to capture the hog.
The key issue I have with this hypothetical is that the Angel doesn’t have any goal other than avoiding the Devil. If that is the Angel’s only goal, then doubling back to a previously available square makes sense as it makes it more difficult for the Devil to box the Angel in. Perhaps I’m just not understanding the hypothetical correctly or perhaps the Angel’s goal is more explicitly laid out in the text.
“The devil theory” You play as the devil and you must trap the angel, you have infinite time, on an infinite plane. The angel moves randomly. The theory is by moving a minimal amount of times to simply fill a 10x10 grid with a space in the middle randomly, after an amount of time he angel would randomly trap herself.
Mathematicians will never stop to amaze me at how they can find the most futile and impossibly useless question in the world and not get bored out of it for decades
In simple terms, "the angel always wins, but if I imply rules I never told you that existed then the angel doesn't"
i'm more confused then i was before. "You'd think the angel can win easily. In fact, angel can win easily."
The angel isn't actually removing squares as it moves, so if it encounters a wall, it can just turn around.
But couldn't the devil just encircle the angel from infinitely far away, such that by the time the angel gets there, they are bound in by a band of spaces with a width that is either equal or higher to k? From there, the devil had confined the movement of the angel, and can just fill in every single block in that area at whatever pace he wants
How is this fascinating? "Here's a problem, and while the answer might seem intuitive, that's because it actually is. I can't/won't explain anything about it, just take my word that something something infinity has this result because...I don't know."
I love how the problem is solved if the angle just didn't move in one direction
The critical thing to consider is that increasing the angel's maximum movement speed necessitates a wall with greater and greater thickness, meaning that while the angel's speed scales with N, the devil's task scales with N^2
Solution: Build a wall and make the angel pay for it.
If the Angel can move 10 tiles per turn, then the devil would need to build the wall so far away that it would take an unreasonably large amount of time to reach it; and he would need to build it on all sides of the angel, making the needed distance 4 times further away than it already was. Then it also needs to be 10 layers thick to prevent movement through it, making the distance needed an extra 10 times on top of that. The Devil could never win in this scenario, because for every tile further across he makes the box, that's an extra turn he needs to spend adding a single block to it in order to get to the other side of the angel and block it in, at which point the angel has already jumped 10 spaces closer to the wall itself which wouldn't be able to hold it. Even if the wall itself was 10 layers thick and the angel couldn't pass it, the devil has no way of adding in an extra side to it which is also 10 layers thick, resulting in the angel escaping and the whole process starting over again. The only one the devil can win is if the angel only has one tile of movement, as building that box around them becomes much easier.