Total Tetris: Tetris with Monominoes, Dominoes, Trominoes, Pentominoes,...

Abstract

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.