Complexity of 2D Snake Cube Puzzles

TL;DR

This work establishes the NP-hardness of several 2D and hexagonal variants of Snake Cube puzzles, including 1×H×W and 2×H×W boxes, as well as hexagonal-prism realizations. The authors develop a unified reduction framework from Numerical 3D Matching, built from block gadgets, wiring lemmas, and a segment-packing principle that force a one-to-one correspondence with a valid 3DM solution. They also address a weak NP-hardness result by encoding repeated patterns with a compressed input form via 2-Partition, and extend the constructions to triangular lattices with a shelf frame to handle multi-layer and closed-chain variants. Collectively, the results deepen our understanding of the algorithmic hardness of puzzle-folding problems and introduce versatile gadgets (segment packing, wires, shelves, and Hamiltonian-fillings) that can be reused across dimensional and geometric variants, with potential implications for puzzle design and complexity theory.

Abstract

Given a chain of $HW$ cubes where each cube is marked "turn $90^\circ$" or "go straight", when can it fold into a $1 \times H \times W$ rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes to be wildcard (can turn or go straight); (2) allowing a larger box with empty spaces (simplifying a proof from CCCG 2022); (3) growing the box (and the number of cubes) to $2 \times H \times W$ (improving a prior 3D result from height $8$ to $2$); (4) with hexagonal prisms rather than cubes, each specified as going straight, turning $60^\circ$, or turning $120^\circ$; and (5) allowing the cubes to be encoded implicitly to compress exponentially large repetitions.

Figures (20)

• Figure 1: An example instance of $D\times H\times W$ Snake Cube with input $D = 1$, $H = 5$, $W = 5$, and the program $\texttt{STTSSSTSSSTTSSTTSSTSTTTTT}$. The gray and white cells represent cubes following instructions $\texttt{T}$ and $\texttt{S}$, respectively.
• Figure 2: The reduction
• Figure 3: Example of the setup wire packing when $n=3$. The red area represents the available space.
• Figure 4: Packing of wires in the reserved spaces on the top and bottom of the rectangle.
• Figure 5: Example of wires after Adjust Crossing step when $X_2$ is placed above $X_1$, $X_3$, and $X_4$. Since wires $U_1$ and $V_1$ are placed before $U_2$ and $V_2$, they do not have to detour around $X_2$. However, since wires $U_3$, $V_3$, $U_4$, and $V_4$ are placed after $U_2$ and $V_2$, they have to detour around $X_2$ (The lengths depicted are very not to scale.)

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: 2-Partition
• Theorem 2.5
• proof
• Definition 2.6
• Theorem 2.7
• proof
• Proposition 2.8