[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-154033-105":59,"doc-detail-154033-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","a-weaker-winning-angel","A Weaker Winning Angel","","In 2006 the angel problem was solved independently by four authors, showing that a sufficiently strong chess piece can escape an adversary that removes one square per move from an infinite chessboard. This work studies a weaker escape for an angel moving on a regular triangular lattice to weakly adjacent triangles. It derives a solution to the original problem from the lattice result, providing a simpler argument and explicitly identifying a weaker winning angel. The analysis proves that an angel starting from the middle square and moving within a 3 by 5 rectangle can win.",{"@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/a-weaker-winning-angel/154033/",{"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/a-weaker-winning-angel/154033.png","ImageObject",300,407,{"name":92,"@type":93},"Rizky","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-17","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the angel problem studied in this document?","Question",{"text":112,"@type":113},"It considers an angel chess piece versus a devil on an infinite board where the devil removes one square each turn and the angel must avoid removed squares. The devil wins if the angel ever has no legal move; otherwise the angel escapes forever.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the triangular lattice result help solve the original problem?",{"text":117,"@type":113},"The paper shows that an angel that moves to weakly adjacent triangles on a regular triangular lattice can escape the devil. A solution to the original infinite-board problem is derived from this lattice escape.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the paper claim about a weaker winning angel?",{"text":121,"@type":113},"Compared with earlier solutions, it proves the existence of a simpler proof and a weaker winning angel. Specifically, it shows that an angel moving from the middle square to any other square of a 3 by 5 rectangle is winning.","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},154033,1787885049,{"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":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},962085564807,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","A weaker winning angel  \nJohan W􀁿astlund 􀀃  \nDepartment of Mathematical Sciences,  \nChalmers University of Technology, S-412 96 G􀁿oteborg, Sweden [wastlund@chalmers.se](wastlund@chalmers.se)  \nMarch 26, 2008  \nAbstract  \nIn 2006 the so-called angel problem was solved independently by four authors. The solution shows that a su􀀎ciently strong chess piece can escape a \\devil\" who removes one square on each move from an in􀀌nite chess board. We show that on the regular triangular lattice, a chess piece that moves to weakly adjacent triangles can escape the devil. A solution to the original problem is derived from this result. Compared to earlier solutions, our approach gives a simpler proof as well as a weaker winning angel. We show that an angel moving from the middle square to any other square of a 3 by 5 rectangle is winning.  \n1 Introduction  \nThe angel problem was introduced by Berlekamp, Conway and Guy [1] in 1982. An angel is a 􀀌nite range chess piece. On an in􀀌nite two-dimensional chess board the angel plays the following game against the devil: On each turn, the devil eats a square of his choice anywhere on the board. Then the angel makes a move. The restriction is that the angel cannot move to a square that has been eaten by the devil. The devil wins if at some point the angel cannot move. Otherwise the angel wins.  \nOnce the legal moves of the angel are speci􀀌ed, we can ask whether it is the angel or the devil who has a winning strategy. Speci􀀌cally, an angel of  \n􀀃 Research supported by the Swedish Research Council  \nFigure 1: An angel of power 3 can move to the shaded squares.  \npower p is a chess piece that can jump in one move to any square at distance at most p king's moves (see Figure 1) .  \nIt was shown in [1] that the devil can trap an angel of power 1, in other words a chess king. This version of the game, also known as \\quadraphage\", had been studied prior to [1] in [5, 7, 14] . The question known as the angel problem, 􀀌rst posed in [1], is whether there is some p such that an angel of power p can win the game.  \nA strange feature of this game is that the angel only wins by escaping forever. One way to obtain a 􀀌nite version of the game is to specify a 􀀌nite region, say an n by n square for some large odd n, let the angel start at the middle, and declare the angel to be the winner if she reaches a square outside this region. If the devil has a winning strategy for some n, then clearly he has a winning strategy for every larger n, and also in the original game. There is a standard compactness argument that shows that the converse is also true. If the angel can win the 􀀌nite game for every n, then since she has only 􀀌nitely many legal moves, there has to be a 􀀌rst move after which she still wins the 􀀌nite game for every n. By consistently choosing such moves she escapes forever, which means she wins the original game. Therefore if  \nthe power p of the angel is speci􀀌ed, then either the angel or the devil has a winning strategy.  \nAnother unusual feature of the game is that however badly the devil plays, he will never be worse than in the initial position. This means that if the devil is winning in the 􀀌rst place, he can never blunder and reach a position where the angel is winning. In fact he can play his 􀀌rst million moves completely at random. If he later discovers a way to trap the angel, those million eaten squares will not harm him. It was pointed out to me by Olle H􀁿aggstr􀁿om that there is an explicit strategy for the devil which is optimal in the sense that if there exist winning strategies for the devil, then this isone of them. Moreover, this strategy is independent of the angel's pattern of movement and therefore wins against any angel that can be beaten! At 􀀌rst this sounds exciting, since it means that in order to exhibit a winning angel itsu􀀎ces to 􀀌nd one that wins against this particular strategy. Unfortunately the strategy for the devil simply consists in trying, for each n, the 􀀌nitely many strat","cbCailJwfgL2LE2A","https://ap.wps.com/l/cbCailJwfgL2LE2A","pdf",531288,17,"English","# Introduction\n## Angel problem and game rules\n## Winning strategies and compactness argument\n## Previous results and weaker angels\n## Power and movement patterns\n## Motivation for weakest winning angel","[{\"question\":\"What is the angel problem studied in this document?\",\"answer\":\"It considers an angel chess piece versus a devil on an infinite board where the devil removes one square each turn and the angel must avoid removed squares. The devil wins if the angel ever has no legal move; otherwise the angel escapes forever.\"},{\"question\":\"How does the triangular lattice result help solve the original problem?\",\"answer\":\"The paper shows that an angel that moves to weakly adjacent triangles on a regular triangular lattice can escape the devil. A solution to the original infinite-board problem is derived from this lattice escape.\"},{\"question\":\"What does the paper claim about a weaker winning angel?\",\"answer\":\"Compared with earlier solutions, it proves the existence of a simpler proof and a weaker winning angel. Specifically, it shows that an angel moving from the middle square to any other square of a 3 by 5 rectangle is winning.\"}]","A Weaker Winning Angel | PDF",43]