[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-154032-105":59,"doc-detail-154032-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","the-angel-and-devil-problem-proof-of-angel-winning-power-2","The Angel and Devil Problem - Proof of Angel Winning Power 2","","A combinatorial game theory problem on an infinite chessboard pits an Angel against a Devil. The Angel moves up to p squares in any direction, while the Devil can burn any squares, blocking the Angel from entering them, with the Devil aiming to trap and the Angel aiming to evade capture. The document analyzes key observations about efficient play, introduces special roles including a “Nice Devil” and a “Runner,” and proves that p=2 is sufficient. It culminates in formal theorems showing the Angel can defeat the Devil with power 2.",{"@graph":69,"@context":122},[70,84,105],{"@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-proof-of-angel-winning-power-2/154032/",{"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-proof-of-angel-winning-power-2/154032.png","ImageObject",300,407,{"name":92,"@type":93},"Dozel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-24","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the rules of the Angel and Devil problem?","Question",{"text":112,"@type":113},"The Angel moves up to p squares in any direction. The Devil burns any chosen squares, preventing the Angel from moving onto burnt squares. The Devil tries to trap the Angel so no legal moves remain, while the Angel tries to evade capture.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why does the paper emphasize the role of reduced paths and prior squares?",{"text":117,"@type":113},"It argues that if the Angel revisits a square it has previously occupied, it achieves nothing and gives the Devil opportunities to burn squares that improve the Devil’s prospects. This motivates focusing on reduced paths that reflect optimal play.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the main result regarding the Angel’s required power?",{"text":121,"@type":113},"The paper proves that p=2 is the minimum power needed for the Angel to win. It presents the Runner and Nice Devil as intermediate constructions, then uses the final proof to establish the decisive outcome for power 2.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},154032,1787885040,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":52,"preview_page_count":4},962085662650,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","The Angel and Devil Problem  \nDhyutidhar Sravankumar  \n27 October 2024  \n1 Introduction  \nThe Angel and Devil problem was a combinatorial game theory problem first proposed by John Conway. It is played on an infinite chessboard with 2 characters. The first is the Angel which can move p squares in any direction for an arbitrary power p that the Angel has. This leads to the Angel’s reach being a square around the Angel with a side length of 2p + 1 . The second character in this game is the Devil. The Devil has an infinite range and can burn any square that it wants. Once a square is burnt by the Devil, the Angel cannot move there. The object of the game for the Devil is to trap the Angel such that the Angel cannot move to any position as they are all burnt. The object of the game for the Angel is to evade capture. This problem was solved in 3 dimensions for a while however that strategy could not be employed to prove the same in the 2 dimensional case. It has also been shown that an Angel of p = 1 loses to the Devil. The objective of the paper is to explain the proof of finding the lowest power which the Angel requires to win. It has been proven that p = 2 is the minimum power needed. In this paper we will explore one of the different solutions presented.  \n2 Observations  \nFirst one could imagine that an Angel which strictly travels in one direction can escape the Devil. In this case we can arbitrarily choose a strictly upward path. This kind of Angel is called a Fool. Although this idea looks intuitive it can easily be countered by the Devil. The Devil must build a wall at a certain range away from the current position of the Fool. The distance from the Fool can be characterized by the equation d = 24p3 . Then after the Angel has traveled ~~1~~2 of the distance from its current point to the wall, the Devil can focus in on the part that the Angel can travel to now. This strategy can beat the Fool for any 1 direction. However, what if the Angel wants to move strictly away from its origin. This way building only 1 wall would not work as the Angel could merely go in the other direction. We could employ a similar strategy however to the one in the beginning where we build 4 walls that entrap the Angel on 4 different sides. This strategy on its own cannot work as the Devil would not have enough  \nFigure 1: The Reduced Path [1]  \ntime to trap the Angel. Rather the Devil must start building its wall 4 times further away. This leads to the equation d = 4 · 24p3 for the distance from the origin on each side. The next observation that must be presented is about the Angel. It must be observed that whenever the Angel moves, if the Angel moves to a square it previously moved to, then it has accomplished nothing while allowing the Devil to burn squares which benefit the Devil. This implies that any square that the Angel has previously moved to is theoretically burnt because moving there would result in worse play for the Angel. This means that every time the Angel makes a move it burns some squares on its own [1] .  \n3 The Nice Devil  \nDefinition 3.1: A Nice Devil is a type of Devil that will not get rid of a square that the Angel has moved to or had the opportunity to move to  \nAssuming that the Devil has a winning strategy, the Nice Devil begins by examining the path and finding a reduced path by finding the oldest square that the Angel can move to (Refer to Figure 2) . It can then either do the exact move that the Devil would do or if it is not available, to pass the turn. It should also be noted that the Devil would not occupy one of the squares that the Angel had moved to in the scenario of the Nice Devil. This is because either the Nice Devil would have occupied it or the path reduction would have proved it wrong [2] .  \nTheorem 3.2: As long as the Devil can trap the Angel with power p in some boundary B (N) then the Nice Devil can do so as well.  \n4 The Runner  \nDefinition 3.3: A Runner is a type of Angel that cannot move over burnt s","cbCaivNaOFHLADWW","https://ap.wps.com/l/cbCaivNaOFHLADWW","pdf",677539,"English","# Introduction\n# Observations\n# The Nice Devil\n# The Runner\n# The Proof\n# Closing Remarks","[{\"question\":\"What are the rules of the Angel and Devil problem?\",\"answer\":\"The Angel moves up to p squares in any direction. The Devil burns any chosen squares, preventing the Angel from moving onto burnt squares. The Devil tries to trap the Angel so no legal moves remain, while the Angel tries to evade capture.\"},{\"question\":\"Why does the paper emphasize the role of reduced paths and prior squares?\",\"answer\":\"It argues that if the Angel revisits a square it has previously occupied, it achieves nothing and gives the Devil opportunities to burn squares that improve the Devil’s prospects. This motivates focusing on reduced paths that reflect optimal play.\"},{\"question\":\"What is the main result regarding the Angel’s required power?\",\"answer\":\"The paper proves that p=2 is the minimum power needed for the Angel to win. It presents the Runner and Nice Devil as intermediate constructions, then uses the final proof to establish the decisive outcome for power 2.\"}]","The Angel and Devil Problem - Proof of Angel Winning Power 2 | PDF"]