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

Abstract

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.Keywords: NP-complete, complexity, puzzles, algorithms IntroductionTetris is one of the most famous puzzle computer games, originally created in 1984 and released in the west on the IBM PC in 1987 * 1 , and substantially popularized by being bundled with every Nintendo Game Boy (except in Japan) [7]. Today, Tetris and its many clones are available to play on almost every platform; the official Tetris mobile game sold over 425 million copies by 2014, eclipsing even the 35 million Game Boy copies [8].The popularity of Tetris has led to many variations, both official and unofficial, with various changes to the rules. Here we consider a theme introduced by Hunter Freyer's Pentris [3], where pieces are pentominoes instead of tetrominoes. Later, Shaunak Kishore's ntris [6] generalized to pieces that are k-ominoes (made of k unit squares joined edge to edge).Our results. In 2004, Breukelaar et al. [1] proved that Tetris is NP-complete for the original tetromino pieces; here we generalize this result to k-omino pieces. More precisely, Breukelaar et al. and we analyze an offline version of Tetris, where the board has a given configuration of occupied squares (resulting from past play, or a complex initial board) and the pieces come from a given sequence of n pieces, and the goals are to survive all the pieces (avoid stacking any piece too high), clear the entire board (remove all occupied squares), or maximize score (according to various measures). While the interesting score functions were clearer for tetrominoes, this aspect has many possible generalizations to andreali@mit.edu f) jaysonl@mit.edu g) ywy@mit.edu Table 1 Complexity results for Tetris with k-ominoes, with or without rotation. New results are marked with a section number ( §). k-ominoes, so we focus on the first two goals. Table 1 summarizes our (and past) results for Tetris with kominoes for the goals of clearing and survival. In addition to the standard rules where pieces can be rotated and translated left/right/down, we consider a variant that forbids rotations. This variation is particularly interesting for dominoes (k = 2), where we can show NP-completeness for clearing, while we conjecture polynomial time with rotation allowed. In total, our results categorize the computational complexity of most variants of Tetris with k-ominoes for most k. With RotationTetris with monominoes (k = 1) is easy (Section 3): we can always survive (by plac...

Tetris is NP-hard even with <i>O</i>(1) Rows or Columns

“…The main open problems are to determine the critical threshold for the minimum number c * of columns and the minimum number r * of rows for which c * -column Tetris and c * -row Tetris are NP-complete, respectively. We proved here that c * ∈ [3,8] and r * ∈ [2,4]. We conjecture that r * = 2, i.e., that 2-row Tetris is NP-complete.…”

“…In Section 6, we solve this problem for the generalization of Tetris to k-omino pieces, denoted k-tris, as implemented in the video games Pentris and ntris, and previously analyzed from a complexity perspective [3]. Specifically, we prove the following results: ( a ) 8-column Empty (≤ 65)-tris is NP-hard.…”

“…[2], Section 2 and Ref. [3], Section 2 respectively for a complete description of the rules. There are in fact many real-world variations of the rules, eventually formalized by The Tetris Company into a modern rule set Ref.…”

“…To make this game easier to analyze from a computational complexity perspective, the perfect-information Tetris problem [2] asks, given an initial board state of filled cells and a sequence of pieces that will arrive, whether the pieces can be played in sequence to either survive (not go above the top row) or clear the entire board (See Section 2 for precise game rules). This problem was proved NP-hard for arbitrary board sizes in 2002 [2], and achesterberg@gmail.com f) jaysonl@mit.edu g) mihirs@mit.edu more recently for the generalization to k-omino pieces for various k [3].…”

On the Complexity of 2-Player Packing Games

“…In polyomino packing, proved hard in [2], the pieces are polyominoes and the board is a rectangle, and "fitting" in this context means being placed with no overlap. Tetris, first proved hard in [4] and analyzed further in [3], uses the same kinds of pieces, but in this puzzle the pieces must be dropped in from the top of a box and fall until they hit other pieces, and the order of the available pieces is a fixed part of the input. Since Robertson and Munro published their analysis of Instant Insanity [7], it has been known that PSPACE-complete problems often take the form of 2-player generalizations of 1-player puzzles.…”