[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-154029-105":59,"doc-detail-154029-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","sneaky-angel-and-devil-game-paper-devil-player-wins-imperfect-information-variant","SNEAKY ANGEL AND DEVIL GAME - Paper - Devil player wins imperfect information variant","","A paper introduces an imperfect-information version of the Angel and Devil game originally proposed in 1982 by Berlekamp, Conway, and Guy. The Devil player has a winning strategy, and the key question is whether this persists when the Angel player receives two moves for each one move by the Devil. The work proves, via its main Theorem 4, that the Devil player still wins the imperfect-information variant. Context is given through determinacy and strategy definitions for abstract games.",{"@graph":69,"@context":123},[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/sneaky-angel-and-devil-game-paper-devil-player-wins-imperfect-information-variant/154029/",{"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/sneaky-angel-and-devil-game-paper-devil-player-wins-imperfect-information-variant/154029.png","ImageObject",300,407,{"name":92,"@type":93},"Putri","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the main focus of the paper on the Angel and Devil game?","Question",{"text":113,"@type":114},"The paper studies an imperfect-information variant where one player’s moves are hidden for several rounds, and investigates whether the Devil’s winning advantage remains under this information restriction.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What key question does the authors’ variant address?",{"text":118,"@type":114},"It asks whether the Devil player’s winning strategy changes when the Angel player is allowed two moves for every one move by the Devil.",{"name":120,"@type":111,"acceptedAnswer":121},"What does Theorem 4 establish?",{"text":122,"@type":114},"Theorem 4 shows that the Devil player wins the imperfect-information variant of the Angel and Devil game.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},154029,1787885022,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"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":130,"read_time":144,"preview_page_count":4},962085571259,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","arXiv :2110 .05485v1 [math .CO] 11 Oct 2021  \nSNEAKY ANGEL AND DEVIL GAME  \nCAMERON BERNSTEIN, LOGAN CRONE, ALEXANDRIA KWON, AND SYDNEY LE  \nAbstract. In this paper we introduce an imperfect information variant of the Angel and Devil game, which was 􀀌rst introduced in 1982 by Berlekamp, Conway, and Guy. The Devil player has a winning strategy in this game, but the main problem they pose is whether this changes if the Angel player is allowed two moves for every one of the Devil. This and many other variants of the game have been considered, and the original question was only solved in 2007, when it was shown that the so-called power 2 Angel could win. Our main theorem, Theorem 4, is that the Devil player wins the imperfect information variant of the game. This generalizes the result that the Devil player has a  \nwinning strategy in the original Angel and Devil game.  \n1. Introduction  \nIn mathematics, a game is characterized by the set of allowed moves and allowed positions that which produces sequences of moves resulting in wins, losses, or draws for each of the players. Most games consist of two players alternating playing various moves. Games can be 􀀌nite, where play ends once a terminal position has been reached, or can be in􀀌nite, with the players producing an in􀀌nite sequence of moves. One can give a formal de􀀌nition of a game in many di􀀋erent ways, and the formal details of the de􀀌nition seldom a􀀋ect the statements of theorems and proofs involved. The main property of interest in studying games is the existence of winning strategies. A strategy is a function which, given a 􀀌nite sequence of moves played so far, will output the next move to play. A winning strategy is a strategy for one of the players that which will always result in a win for that player. A game is then said to be determined if one of the players has a winning strategy.  \nGames of perfect information are those in which all players have full knowledge of the rules and payo􀀋, as well as a complete list of all moves played by both players in each round. In games of imperfect information, some information is not revealed to the players. In most games of imperfect information, it is the opponent’s moves that are hidden. Games with imperfect information are more likely to mirror interactions in reality, but are also more di􀀎cult to analyze, as the notion of a strategy must be adapted to this context as well. The only type of imperfect information game we consider here is one in which one of the player’s moves are hidden for some number of rounds from the other player. For our purposes, a strategy in this game will be playing deterministically (not probabilistically) against only the moves which have been revealed. So for such a strategy to be a winning strategy, it must win against all possible moves by the opponent.  \nKey words and phrases. Games, Combinatorics.  \nThe authors would like to thank the Math Department of the University of North Texas’s Incubator project for its encouragement and guidance.  \n2 CAMERON BERNSTEIN, LOGAN CRONE, ALEXANDRIA KWON, AND SYDNEY LE  \nIn 1982, Berlekamp, Conway, and Guy [1] introduced a game of perfect information in which one of the players is moving a piece on an in􀀌nite chess board, winning only if they are able to keep moving forever, while the other player opposes the 􀀌rst by damaging the board and attempting to trap their piece. They posed a problem regarding which player has a winning strategy in the case that the piece is allowed to make two chess-king moves every round. This problem stood unsolved for many years, with Conway [3] revisiting the problem and Winkler [8] popularizing it. Some progress was made by Kutz [5] in 2004, with the problem being fully answered in 2007 by Bowditch [2] and M􀀓ath􀀓e [7] .  \nIf a game of perfect information is su􀀎ciently simple, then it is known that oneof the players must have a winning strategy and is thus said to be determined, and the abstract study of which games are determined","cbCaiadvEws0DAbk","https://ap.wps.com/l/cbCaiadvEws0DAbk","pdf",150282,"English","# Introduction\n## Games and winning strategies\n# Definition of the game\n## Angel and Devil game\n# Main results\n## Restricted Angels and strategy sequences\n## Adapting strategies to general Angels","[{\"question\":\"What is the main focus of the paper on the Angel and Devil game?\",\"answer\":\"The paper studies an imperfect-information variant where one player’s moves are hidden for several rounds, and investigates whether the Devil’s winning advantage remains under this information restriction.\"},{\"question\":\"What key question does the authors’ variant address?\",\"answer\":\"It asks whether the Devil player’s winning strategy changes when the Angel player is allowed two moves for every one move by the Devil.\"},{\"question\":\"What does Theorem 4 establish?\",\"answer\":\"Theorem 4 shows that the Devil player wins the imperfect-information variant of the Angel and Devil game.\"}]","SNEAKY ANGEL AND DEVIL GAME - Paper - Devil player wins imperfect information variant | PDF",25]