1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete

Authors Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, Avi Zeff

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Josh Brunner
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Lily Chung
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Dylan Hendrickson
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Adam Hesterberg
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Adam Suhl
  • Algorand, Boston, MA, USA
Avi Zeff
  • Massachusetts Institute of Technology, Cambridge, MA, USA


We thank the many colleagues over the years for their early collaborations in trying to resolve the 1 × 1 Rush Hour problem (when E. Demaine mentioned it to various groups over the years): Timothy Abbott, Kunal Agrawal, Reid Barton, Punyashloka Biswal, Cy Chen, Martin Demaine, Jeremy Fineman, Seth Gilbert, David Glasser, Flena Guisoresac, MohammadTaghi Hajiaghayi, Nick Harvey, Takehiro Ito, Tali Kaufman, Charles Leiserson, Petar Maymounkov, Joseph Mitchell, Edya Ladan Mozes, Krzysztof Onak, Mihai Pǎtraşcu, Guy Rothblum, Diane Souvaine, Grant Wang, Oren Weimann, Zhong You (MIT, November 2005); Jeffrey Bosboom, Sarah Eisenstat, Jayson Lynch, and Mikhail Rudoy (MIT 6.890, Fall 2014); and Joshua Ani, Erick Friis, Jonathan Gabor, Josh Gruenstein, Linus Hamilton, Lior Hirschfeld, Jayson Lynch, John Strang, Julian Wellman (MIT 6.892, Spring 2019, together with the present authors).

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Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff. 1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Consider n²-1 unit-square blocks in an n × 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 × 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 × 2 and horizontal 2 × 1 movable blocks and 4-color Subway Shuffle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • puzzles
  • sliding blocks
  • PSPACE-hardness


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