{"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/1-x-1-rush-hour-with-fixed-blocks-is-pspace","title":"1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete","arxiv_id":"2003.09914","date":"2020-03-22","proceeding":null,"authors":["Josh Brunner","Lily Chung","Erik D. Demaine","Dylan Hendrickson","Adam Hesterberg","Adam Suhl","Avi Zeff"],"abstract":"Consider $n^2-1$ unit-square blocks in an $n \\times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 \\times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 \\times 2$ and horizontal $2 \\times 1$ movable blocks and 4-color Subway Shuffle.","url_abs":"https://arxiv.org/abs/2003.09914v2","url_pdf":"https://arxiv.org/pdf/2003.09914v2.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":"1-x-1-rush-hour-with-fixed-blocks-is-pspace","repo_url":"https://github.com/edemaine/svgtiler","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}