**Reference**:- Erik D. Demaine, Martin L. Demaine, and Joseph O'Rourke, “PushPush and Push-1 are NP-hard in 2D”, in
*Proceedings of the 12th Annual Canadian Conference on Computational Geometry (CCCG 2000)*, Fredericton, New Brunswick, Canada, August 16–18, 2000, pages 211–219. **Abstract**:-
We prove that two pushing-blocks puzzles are intractable in 2D.
One of our constructions improves
an earlier result that established intractability in
3D [OS99] for
a puzzle inspired by the game
*PushPush*. The second construction answers a question we raised in [DDO00] for a variant we call*Push-1*. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a block is pushed it slides the maximal extent of its free range. In the Push-1 version, the agent can only push one block one square at a time, the minimal extent—one square. Both NP-hardness proofs are by reduction from SAT, and rely on a common construction. **Comments**:- This paper is also available from the electronic proceedings as http://www.cs.unb.ca/conf/cccg/eProceedings/26.ps.gz, and as cs.CG/0007021 of the Computing Research Repository (CoRR).
This version is updated from the original conference version in order to fix an error discovered by and solved together with Thomas Shermer.

**Length**:- The paper is 9 pages.
**Availability**:- The paper is available in PostScript (760k) and gzipped PostScript (206k).
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**Related papers**:- PushPush2DTR (PushPush is NP-hard in 2D)
**Related webpages**:- Pushing Blocks

See also other papers by Erik Demaine.

Last updated January 13, 2020 by Erik Demaine.