The Three-Color and Two-Color TantrixTM Rotation Puzzle Problems Are NP-Complete Via Parsimonious Reductions

Holzer and Holzer [7] proved that the TantrixTM 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 TantrixTM 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 TantrixTM 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 [7] 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. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.