{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/complexity-of-retrograde-and-helpmate-chess","title":"Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard","arxiv_id":"2010.09271","date":"2020-10-19","proceeding":null,"authors":["Josh Brunner","Erik D. Demaine","Dylan Hendrickson","Julian Wellman"],"abstract":"We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A \"retrograde\" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is \"valid\" or \"legal\" or \"reachable\". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A \"helpmate\" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.","url_abs":"http://arxiv.org/abs/2010.09271v1","url_pdf":"http://arxiv.org/pdf/2010.09271v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"complexity-of-retrograde-and-helpmate-chess","repo_url":"https://github.com/tromp/ChessPositionRanking","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}