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It defines the k-angel’s movement rules on an infinite chessboard and the Devil’s removal strategy that limits future moves. Winning conditions are characterized by whether the Angel can move indefinitely without being trapped and whether the Devil can confine the k-angel within a corridor of width 2k+1. The discussion surveys basic propositions and greedy approaches, explaining why intuitive strategies fail and presenting a winning method against specific angel behaviors.",{"@graph":69,"@context":127},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-angel-and-devil-problem-winning-strategy-and-strategies-that-dont-work/154028/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-angel-and-devil-problem-winning-strategy-and-strategies-that-dont-work/154028.png","ImageObject",300,407,{"name":92,"@type":93},"Bintang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-07","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119,123],{"name":110,"@type":111,"acceptedAnswer":112},"What is the main goal of the Angel and Devil Problem?","Question",{"text":113,"@type":114},"Determine whether there exists a k such that a k-angel can win the game, or more generally which k allow winning for which player.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does a k-angel move and what does the Devil do each turn?",{"text":118,"@type":114},"The angel moves first by up to k in each coordinate to a point that is not on the removed squares, while the Devil removes a square each turn (not the angel’s current square) so the angel can’t land on it later, though it can fly over it.",{"name":120,"@type":111,"acceptedAnswer":121},"Why do the article’s greedy potential-function strategies fail?",{"text":122,"@type":114},"The Devil exploits the potential function’s sensitivity by shaping traps so that greedy minimizing moves lure the angel into a confined region, with trap geometry chosen to counter the function’s behavior.",{"name":124,"@type":111,"acceptedAnswer":125},"What happens when the angel uses the “fool” strategy?",{"text":126,"@type":114},"The article states that the Devil always wins against a fool, since the fool’s positions remain inside a bounded upward cone, allowing the Devil to remove reachable segments step by step until the angel can no longer avoid confinement.","https://schema.org",{"og:url":83,"og:type":129,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":131,"canonical":83},"index,follow",{"doc_id":133,"site_id":62},154028,1787885016,{"code":4,"msg":5,"data":136},{"doc_id":133,"user_id":137,"nickname":92,"user_avatar":138,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":139,"file_id":140,"file_url":141,"file_type":142,"file_size":143,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":144,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":145,"faqs":146,"seo_title":147,"seo_description":67,"update_tm":134,"read_time":148,"preview_page_count":4},962085564381,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","THE ANGEL AND DEVIL PROBLEM  \nARJUN KULKARNI  \nAbstract. This article will discuss the The Angel and Devil Problem in the subject of combinatorial game theory. We will go over the game, the problem, and then discuss a winning strategy and strategies that appear to win, but don't.  \n1. Introduction  \nWe will begin by giving an overview of the Angel and Devil Game, and then going over the Angel and Devil Problem. The Angel and Devil Game is a combinatorial game involving 2 players, one of them being the Angel, and one of them being the Devil, on an in􀀌nite chessboard. The Angel will move 􀀌rst, and will move for some k from (x; y) to any point (x0 ; y0 ) such that (x0 ; y0 )  (x; y) and jx0 􀀀 xj 􀀔 k , jy0 􀀀 yj 􀀔 k. k is called the power of the angel, and an angel with a power of k will be called a k-angel. The Devil will move second, and each turn will take a square on the board and remove it, so that an angel can no longer move to that square, but can move over that square because the Angel is 􀀍ying. The only restriction on which square is removed is that it can't be the square the angel is currently standing on. In order to win as the Angel, one must be able to move inde􀀌nitely, and never be trapped by the Devil. In order to win as the Devil, one must trap the k-angel in a square of width 2k + 1 . The Angel and Devil Problem reads, is there a k such that a k-angel can win the game? In this article, we will discuss this problem and the general problem of which k's will win for which players.  \n2. Basic Strategies  \nWe will begin with the following basic, but important observation.  \nProposition 2.1 . If a k-angel can win, then a k0-angel can also win, for any k0 􀀕 k.  \nProof. We will give an informal proof to this observation. First, if the angel will not get trapped in a square of width 2k + 1, then it will not get trapped in a square of greater width because the amount of extra squares needed to be removed gives the Angel enough time to escape the square, especially considering the increased maneuverability of the angel.  \n􀀄  \nNow we will move onto basic strategies that seem to work, but do not actually.  \nAlgorithm 2.2 . We will create a potential function which will change depending on squares already removed. Every move will greedily go to a square which minimizes the potential function.  \nThe reason this general algorithm does not work is because the Devil will always 􀀌nd away to exploit the strategy by 􀀌nding features of the function which make it sensitive to certain removed blocks. We will show this with an example.  \nDate: August 24, 2021 .  \n2 ARJUN KULKARNI  \nExample. The potential function will increase most by squares near the angel, but not so much by squares farther away from the angel.  \nThis example is of a potential function sensitive to squares near it. So the devil can exploit the sensitive squares with a trap. The trap will be a large amount of removed squares in the shape of horseshoe with a big opening, and will be extremely far away from the angel. At any time that we want, we can locate the angel, and remove a square extremely close toit. In order to minimize the potential function, the angel will move in the exact opposite direction of the square we just removed. So we can take advantage of this and lure the angel into the horseshoe trap, and then close o􀀋 the horseshoe. We can always change the size of the horseshoe to make this work. So this is one example of taking advantage of the function's sensitive squares.  \nExample. The potential function will increase most by squares very far away from it, but not so much by squares near it.  \nFor this example, we can take advantage of squares far away from the angel, and create a horseshoe trap near the angel. We can follow the same strategy as last time by putting squares far away from the angel and luring it into the horseshoe trap, which doesn't e􀀋ect the potential function that much since it is extremely close to the angel.  \nIn general, this greedy stra","cbCaikKJBie1y6kC","https://ap.wps.com/l/cbCaikKJBie1y6kC","pdf",252808,"English","# Introduction\n## Angel and Devil Game Rules\n## Angel and Devil Problem\n# Basic Strategies\n## Proposition 2.1\n## Greedy Potential Function Strategies\n## Fool Angels and Theorem 2.4","[{\"question\":\"What is the main goal of the Angel and Devil Problem?\",\"answer\":\"Determine whether there exists a k such that a k-angel can win the game, or more generally which k allow winning for which player.\"},{\"question\":\"How does a k-angel move and what does the Devil do each turn?\",\"answer\":\"The angel moves first by up to k in each coordinate to a point that is not on the removed squares, while the Devil removes a square each turn (not the angel’s current square) so the angel can’t land on it later, though it can fly over it.\"},{\"question\":\"Why do the article’s greedy potential-function strategies fail?\",\"answer\":\"The Devil exploits the potential function’s sensitivity by shaping traps so that greedy minimizing moves lure the angel into a confined region, with trap geometry chosen to counter the function’s behavior.\"},{\"question\":\"What happens when the angel uses the “fool” strategy?\",\"answer\":\"The article states that the Devil always wins against a fool, since the fool’s positions remain inside a bounded upward cone, allowing the Devil to remove reachable segments step by step until the angel can no longer avoid confinement.\"}]","The Angel and Devil Problem - Winning Strategy and Strategies That Don't Work | PDF",13]