We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH+04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger $k$-omino pieces is NP-complete even when the board starts empty, even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.

中文翻译：

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即使有 $O(1)$ 行或列，俄罗斯方块也是 NP-hard

我们证明，即使限制为 8 列或 4 行，经典的落块视频游戏俄罗斯方块（生存和棋盘清理）仍然是 NP 完全的，解决了 15 年前提出的开放问题 [BDH+04]。我们的减少来自 3-Partition，类似于之前对不受限制的电路板尺寸的减少，但具有更好的桶包装。从积极的方面来说，我们证明了 2 列俄罗斯方块（和 1 行俄罗斯方块）是多项式。我们还证明，即使棋盘开始为空，即使限制为 3 列或 2 行或恒定大小的棋子，俄罗斯方块对更大的 $k$-omino 棋子的泛化也是 NP 完全的。最后，我们展示了一个动画俄罗斯方块字体。