Tetris is Hard with Just One Piece Type

TL;DR

It is proved, for any tetromino piece type $P$ except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of $P$ pieces.

Abstract

We analyze the computational complexity of Tetris clearing (determining whether the player can clear an initial board using a given sequence of pieces) and survival (determining whether the player can avoid losing before placing all the given pieces in an initial board) when restricted to a single polyomino piece type. We prove, for any tetromino piece type $P$ except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of $P$ pieces. These surprising results disprove a 23-year-old conjecture on the computational complexity of Tetris with only I pieces (although our result is only for a specific rotation system). As a corollary, we prove the NP-hardness of Tetris clearing when the sequence of pieces has to be able to be generated from a $7k$-bag randomizer for any positive integer $k\geq 1$. On the positive side, we give polynomial-time algorithms for Tetris clearing and survival when the input sequence consists of only dominoes, assuming a particular rotation model, solving a version of a 9-year-old open problem. Along the way, we give polynomial-time algorithms for Tetris clearing and survival with $1\times k$ pieces (for any fixed $k$), provided the top $k-1$ rows are initially empty, showing that our I NP-hardness result needs to have filled cells in the top three rows.

Figures (83)

• 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 $90^\circ$ counterclockwise. 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 3: Falling rotation model for dominoes: when the domino on the left attempts to rotate clockwise or counterclockwise, it becomes the respective domino on the right, if that position is unobstructed.
• Figure 4: Procedure for clearing holes for $k=3$. The dots in (a) show which empty squares are holes.
• Figure 5: Path order of empty cells in an example board state. Here, a bigger number appears later in the path order.

Theorems & Definitions (22)

• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof