Tetris with Few Piece Types

TL;DR

This work determines the boundary between easy and hard instances of Tetris under the Super Rotation System by showing that for every $2$-element subset of tetromino types, Tetris clearing is $NP$-hard and the counting version is $#P$-hard, with ASP-completeness for certain three-piece sets. The authors develop a versatile bottle-and-finisher reduction framework from distinct instances of 3-Partition and Numerical 3-Dimensional Matching with Distinct Integers, and they extend hardness results to restricted gravity models (hard drops and $20G$) for small piece sets. They also establish $ASP$-completeness for specific $3$-piece subsets and discuss survival variants, clarifying the impact of piece-choice on problem difficulty. Overall, the paper resolves a long-standing open problem about which piece sets induce hardness in Tetris under SRS and provides a broad methodology for complexity proofs in combinatorial games. The results have implications for understanding computational hardness in modern Tetris variants and related packing/tiling problems, and they open several avenues for future work on single-piece-type cases and alternative randomization schemes.

Abstract

We prove NP-hardness and #P-hardness of Tetris clearing (clearing an initial board using a given sequence of pieces) with the Super Rotation System (SRS), even when the pieces are limited to any two of the seven Tetris piece types. This result is the first advance on a question posed twenty years ago: which piece sets are easy vs. hard? All previous Tetris NP-hardness proofs used five of the seven piece types. We also prove ASP-completeness of Tetris clearing, using three piece types, as well as versions of 3-Partition and Numerical 3-Dimensional Matching where all input integers are distinct. Finally, we prove NP-hardness of Tetris survival and clearing under the "hard drops only" and "20G" modes, using two piece types, improving on a previous "hard drops only" result that used five piece types.

Figures (48)

• Figure 1: All tetromino pieces, in order from top to bottom: $I$, $J$, $L$, $O$, $S$, $T$, $Z$. The first column is the default orientation of a piece upon spawning in; each column to the right indicates a $90^\circ$ rotation clockwise about the rotation center of the piece.
• Figure 2: An example of the SRS kick system. Suppose the $J$ piece in (a) is being rotated $90^\circ$ counter-clockwise. Test 1 (which is $(0, 0)$) would fail, due to the dark gray square shown in (b). Test 2 (which is $(+1, 0)$) would succeed, as shown in (c), and so the $J$ piece would rotate to the position in (c).
• Figure 4: The $J$ and $T$ finishers (the $L$ finisher can be obtained by reflecting the $J$ finisher through a vertical line), and an example of the $J$ finisher next to a bottle (in the ${O, J}$ setup).
• Figure 5: The bottle structure for $\{I, L\}$. (b) shows how an $L$ piece must block a bottle during the priming sequence. (c) shows the result of a filling sequence where $a_i=2$. (d) shows where the $L$ piece in the closing sequence must go. (e) shows the result of the closing sequence.
• Figure 6: The bottle structures for the other subsets containing $I$, including how the non-$I$ piece must block a bottle during the priming and closing sequence.

Theorems & Definitions (31)

• Theorem 3.3
• proof : Proof of Theorem \ref{['3partasp']}
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition 4.4
• proof