Tetris is NP-hard even with $O(1)$ rows or columns

TL;DR

This paper proves that Tetris remains NP-complete even with very limited board sizes, but is polynomial for 2-column or 1-row versions, and extends NP-completeness to larger pieces and restricted boards.

Contribution

It establishes NP-hardness of Tetris with 8 columns or 4 rows, resolving longstanding open problems, and shows polynomial solvability for 2-column and 1-row cases, also analyzing larger pieces.

Findings

Tetris is NP-complete with 8 columns or 4 rows.

2-column and 1-row Tetris are polynomial.

Larger k-omino Tetris is NP-complete even with small boards.

Abstract

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.