Tetris is Hard, Even to Approximate
In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p 1−ɛ , when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2−ε, for any ε > 0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piecesets.
KeywordsRotation Model Staging Area Instantaneous Model Rectilinear Polygon Original Piece
Unable to display preview. Download preview PDF.
- 1.R. Breukelaar, H. J. Hoogeboom, and W. A. Kosters. Tetris is hard, made easy. Technical report, Leiden Institute of Advanced Computer Science, 2003.Google Scholar
- 2.J. Brzustowski. Can you win at Tetris? Master’s thesis, U. British Columbia, 1992.Google Scholar
- 3.H. Burgiel. How to lose at Tetris. Mathematical Gazette, July 1997.Google Scholar
- 4.E. D. Demaine. Playing games with algorithms: Algorithmic combinatorial game theory. In Proc. MFCS, pages 18–32, August 2001. cs.CC/0106019.Google Scholar
- 5.E. D. Demaine, S. Hohenberger, and D. Liben-Nowell. Tetris is hard, even to approximate. Technical Report MIT-LCS-TR-865, 2002. cc.CC/0210020.Google Scholar
- 9.S. Kim. Tetris unplugged. Games Magazine, pages 66–67, July 2002.Google Scholar
- 10.M. M. Kostreva and R. Hartman. Multiple objective solution for Tetris. Technical Report 670, Department of Mathematical Sciences, Clemson U., May 1999.Google Scholar
- 12.D. Sheff. Game Over: Nintendo’s Battle to Dominate an Industry. Hodder and Stoughton, London, 1993.Google Scholar
- 13.Tetris, Inc. http://www.tetris.com.Google Scholar
- 14.U. Zwick. Jenga. In Proc. SODA, pages 243–246, 2002.Google Scholar