Total Tetris: Tetris with Monominoes, Dominoes, Trominoes, Pentominoes,...
Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Adam Hesterberg, Andrea Lincoln, Jayson Lynch, Yun William Yu · Journal of Information Processing · 2017
We consider variations on the classic video game Tetris where pieces are k-ominoes instead of the usual tetrominoes (k=4), as popularized by the video games ntris and Pentris. We prove that it is NP-complete to survive or clear a given initial board with a given sequence of pieces for each k ≥ 5, complementing the previous NP-completeness result for k=4. More surprisingly, we show that board clearing is NP-complete for k=3; and if pieces may not be rotated, then clearing is NP-complete for k=2 and survival is NP-complete for k=3. All of these problems can be solved in polynomial time for k=1.