2007/11/12 by Dorothea Baumeister, Baumeister, Dorothea, Joerg Rothe +1
Computer Science · #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #F.1.3 #F.2.2 #FOS: Computer and information sciences #cs.CC
paper · pdf · doi:10.48550/arxiv.0711.1827
30 pages, 25 figures
openalex publication_date 2007/11/12 · arxiv created 2008/06/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix(TM) rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix(TM) tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.