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arXiv · 2603.09958

Tetris is Hard with Just One Piece Type

MIT Hardness Group·Josh Brunner·Erik D. Demaine·Della Hendrickson·Jeffery Li
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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.

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MIT Hardness Group, Josh Brunner, Erik D. Demaine, Della Hendrickson, Jeffery Li. 2026-03-10. Tetris is Hard with Just One Piece Type. https://arxiv.org/abs/2603.09958

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