Tetris: Complexity, AI, and Formal Models

Updated 7 July 2026

Tetris is a puzzle game defined by its falling tetromino pieces, board-clearing mechanics, and strategic placement that delays termination.
Research uses Tetris as a testbed for computational complexity, reinforcement learning, and formal algebraic modeling through diverse game variants.
Algorithmic studies reveal that handcrafted feature engineering and state representation are crucial for tackling NP-hard decision making in Tetris.

Searching arXiv for recent and foundational Tetris papers relevant to computational complexity, AI, and formal models. arXiv search results for query "Tetris" (showing the most relevant and papers):

(Chen et al., 24 Mar 2026) — "Bitboard version of Tetris AI"
(Wang et al., 19 Mar 2026) — "Evaluating Game Difficulty in Tetris Block Puzzle"
(Group et al., 10 Mar 2026) — "Tetris is Hard with Just One Piece Type"
(Wang, 2 Mar 2026) — "Diffusion-MPC in Discrete Domains: Feasibility Constraints, Horizon Effects, and Critic Alignment: Case study with Tetris"
(Group et al., 2024) — "Tetris with Few Piece Types"
(Asif et al., 2020) — "Tetris is NP-hard even with O(1) rows or columns"
(Jentsch et al., 2020) — "Exploring Tetris as a Transformation Semigroup"
(Algorta et al., 2019) — "The Game of Tetris in Machine Learning"
0210020 — "Tetris is Hard, Even to Approximate" Tetris is a popular puzzle video game, invented in 1984, in which the player places a sequence of tetromino pieces into a rectangular gameboard; any completely filled row is cleared and all pieces above it drop by one row. In the research literature, Tetris functions simultaneously as a canonical combinatorial game, a benchmark for sequential decision making under uncertainty, and a testbed for questions in computational complexity, reinforcement learning, planning, and formal algebraic modeling (Jentsch et al., 2020, Algorta et al., 2019).

1. Game structure and formal models

In the standard academic formulation, Tetris is played on a $20\times 10$ board with the 7 tetrominoes. A state is typically defined as the current board configuration together with the identity of the falling piece, while an action is a legal final placement reachable by rotating and translating the current piece before dropping it (Algorta et al., 2019, Wang, 2 Mar 2026). Some variants additionally expose the next piece, and some studies change the movement model, rotation system, or scoring rule, but the dominant formalization remains a Markov decision process with state $=$ board $+$ current piece and action $=$ legal placement (Algorta et al., 2019).

The game’s compact rules conceal an unusually large search space. One estimate reported in the machine-learning literature is about $7 \times 2^{200}$ states, reflecting the binary occupancy of the board together with the current tetromino identity (Algorta et al., 2019). The same literature also emphasizes a structural fact that shapes both theory and algorithm design: every game terminates with probability $1$, because some piece sequences force termination regardless of play (Algorta et al., 2019). This places Tetris in a distinctive regime: it is not an infinite-horizon control problem with a truly survivable optimal policy, but a finite-yet-unbounded sequential optimization problem dominated by delayed consequences.

A further formal distinction concerns “state” versus “afterstate”. In many Tetris controllers, especially placement-based ones, the essential object is the board after placing the current piece and clearing lines, before the next random piece appears. This afterstate decomposition is central to later reinforcement-learning work because the geometric effect of a move is deterministic once the current piece and placement are fixed; only the next-piece draw is stochastic (Chen et al., 24 Mar 2026).

2. Strategic regularities and evaluation criteria

Strong Tetris play is usually described not in terms of raw cells but through structured board features. The review literature identifies a long-running consensus around holes, transitions, wells, height, and related abstractions as the most informative descriptors of board quality (Algorta et al., 2019). Pierre Dellacherie’s controller became the canonical example of this feature-based view. It used six features—number of holes, landing height, row transitions, column transitions, cumulative wells, and eroded cells—with the linear evaluation function

$- 4 \times holes - cumulative\:wells - row\:transitions - column\:transitions - landing\:height + eroded\:cells$

and reportedly cleared about 660,000 lines on the full $20\times 10$ grid (Algorta et al., 2019).

This line of work established a durable empirical lesson: representation mattered more than generic optimization. The dominant controller architecture in the literature scores each legal placement with a linear evaluation function over state or state–action features, then selects the highest-valued move (Algorta et al., 2019). A related observation is that many candidate placements are qualitatively dominated before exact weights are even known. Under simple dominance and cumulative dominance, the median number of candidate placements was reported to fall from 17 to 3 and then to 1, reframing part of Tetris decision making as structured elimination rather than pure value estimation (Algorta et al., 2019).

The same literature also marks an important limitation. Existing solutions that survive for millions of lines under standard one-line scoring are explicitly said to remain far short of expert human play when humans are allowed to play without time pressure (Algorta et al., 2019). This suggests that high-score play under simplified reward functions should not be conflated with full strategic mastery. A plausible implication is that Tetris remains difficult not only because of local packing, but because of long-horizon subgoal formation, delayed payoffs, and the need for board abstractions that current systems still obtain largely by hand.

3. Classical computational complexity

The theoretical literature treats offline Tetris—where the entire piece sequence is known in advance—as a family of decision and optimization problems. A foundational result shows that, in this offline version, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises, minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends [0210020]. The same paper proves strong inapproximability: maximizing cleared rows and maximizing placed pieces are inapproximable within a factor of $p^{1-\epsilon}$ when the input contains $p$ pieces, and minimizing maximum height is inapproximable within a factor of $=$ 0, for any $=$ 1 [0210020].

Later work showed that this hardness is not an artifact of large boards. Classic Tetris remains NP-complete, for both survival and board clearing, even when restricted to 8 columns or to 4 rows (Asif et al., 2020). At the same time, the smallest extremes admit efficient algorithms: 2-column Tetris and 1-row Tetris are polynomial-time solvable (Asif et al., 2020). This sharply narrows the geometric boundary between easy and hard instances. The same paper extends hardness to generalized $=$ 2-omino Tetris, proving NP-completeness even when the board starts empty, even when restricted to 3 columns or 2 rows, and even when all pieces have constant size (Asif et al., 2020).

These results correct a common misconception that Tetris is hard only because the standard board is wide or because exact optimization requires all seven pieces interacting on large instances. The complexity-theoretic picture is substantially stronger: the game remains intractable under severe geometric restrictions.

4. Piece-restricted and rule-restricted hardness

A second misconception is that hardness depends on having the full tetromino alphabet. That is false under modern rotation rules. Under the Super Rotation System, Tetris clearing is NP-hard and $=$ 3-hard even when the available pieces are restricted to any two of the seven tetromino types (Group et al., 2024). The same work proves ASP-completeness of Tetris clearing using three piece types, and also proves NP-hardness of Tetris survival and clearing under hard-drops-only and 20G modes using only two piece types (Group et al., 2024). This was the first substantive progress on the long-standing classification question of which restricted piece sets are easy versus hard.

The boundary was pushed further in 2026. For any tetromino type $=$ 4 except $=$ 5, Tetris clearing and survival under SRS are NP-hard even when the input sequence consists only of copies of $=$ 6 (Group et al., 10 Mar 2026). Thus $=$ 7-only, $=$ 8-only, $=$ 9-only, $+$ 0-only, $+$ 1-only, and $+$ 2-only Tetris are all hard. As a corollary, Tetris clearing remains NP-hard when the sequence must be generable from a $+$ 3-bag randomizer for any positive integer $+$ 4 (Group et al., 10 Mar 2026). This result explicitly disproves a 23-year-old conjecture about the complexity of $+$ 5-only Tetris, at least under SRS (Group et al., 10 Mar 2026).

The same paper also identifies tractable islands. It gives polynomial-time algorithms for Tetris clearing and survival with dominoes under a falling rotation model, and more generally for $+$ 6 pieces when the top $+$ 7 rows are initially empty (Group et al., 10 Mar 2026). For $+$ 8, this shows that the $+$ 9-piece hardness result depends essentially on the combination of SRS and nonempty top rows. The positive and negative results therefore do not merely classify piece sets; they isolate the computational role of the rotation model and upper-board geometry.

5. Tetris as an AI benchmark

The modern AI literature treats Tetris as an unresolved benchmark for feature discovery, sample-efficient reinforcement learning, autonomous option learning, and long-horizon planning (Algorta et al., 2019). Early approximate dynamic-programming methods were modest: Tsitsiklis and Van Roy obtained around 30 cleared lines on a $=$ 0 grid using only holes and maximum height; Bertsekas and Tsitsiklis reached around 2,800 lines with a richer feature set; Lagoudakis et al. reported roughly 1,000–3,000 lines; Kakade’s natural policy gradient reached about 6,800 lines; and Farias and Van Roy reached about 4,500 lines with linear programming and sampled Bellman constraints (Algorta et al., 2019).

Performance later shifted decisively toward feature-engineered controllers with better optimization. Böhm et al. reported 480,000,000 lines with a controller that used both the current and next piece; Szita and Lőrincz reached about 350,000 lines with the cross-entropy method; the BCTS controller reached around 35,000,000 cleared lines; and Gabillon et al.’s classification-based modified policy iteration reached about 51,000,000 lines, the strongest result covered in the review (Algorta et al., 2019). Yet the same review stresses that these successes still depend heavily on handcrafted abstractions and do not amount to autonomous representation learning from raw board inputs (Algorta et al., 2019).

Recent work uses Tetris as a diagnostic domain for newer planning paradigms. A diffusion-based model-predictive controller showed that feasibility masking is essential in discrete combinatorial settings, that naive DQN reranking is strongly misaligned with rollout quality, and that shorter planning horizons can outperform longer ones under sparse and delayed rewards (Wang, 2 Mar 2026). Systems work has also focused on the benchmark infrastructure itself: a bitboard-based Tetris engine reported a 53-fold speedup over OpenAI Gym-Tetris, an afterstate-evaluating actor network that outperformed a traditional action-value actor with fewer parameters, and a buffer-optimized PPO that achieved an average score of 3,829 on $=$ 1 grids within 3 minutes (Chen et al., 24 Mar 2026). These results do not settle Tetris, but they clarify where progress is bottlenecked: efficient simulation, afterstate representation, feasibility constraints, and long-horizon credit assignment.

Beyond complexity and machine learning, Tetris has also been studied as a formal algebraic object. One 2020 abstract formulates two versions of the game as a transformation semigroup and analyzes them through Krohn–Rhodes theory; in a restart-on-loss variation, it reports permutation group structures including the symmetric group $=$ 2, implying, at least in a simple case, that iterated Tetris is finitarily computationally universal (Jentsch et al., 2020). This places Tetris in conversation not only with search and optimization, but also with semigroup decomposition and finite-state computation.

The literature also contains nearby, but distinct, Tetris-derived puzzle domains. “Tetris Block Puzzle”, for example, is an $=$ 3 single-player stochastic placement game without the classic falling-block dynamics. In that setting, increasing the number of holding blocks $=$ 4 and preview blocks $=$ 5 reduces difficulty, while adding more block variants increases difficulty; among the tested additions, the T-pentomino produced the largest slowdown for a Stochastic Gumbel AlphaZero evaluator (Wang et al., 19 Mar 2026). This should not be confused with classic Tetris, but it demonstrates how the broader Tetris family serves as a laboratory for rule-variation studies.

Taken together, these research directions support a broad characterization. Tetris is not merely a popular puzzle game; it is a compact formal system in which local geometric actions generate deep questions about intractability, representation, planning under uncertainty, action abstraction, and algebraic structure. The persistence of hardness under tiny boards, tiny piece alphabets, and even single-piece regimes, together with the continued reliance of strong agents on carefully engineered abstractions, explains why Tetris remains a durable object of theoretical and algorithmic study 0210020.