Complexity of a Tetris variant

The game complexity of Tetris was first studied by Erik demaine, Susan Hohenberger and David LibenNowell in “Tetris is hard" [1]. Their proof was later simplified in “Tetris is hard, made easy” [2]. The authors of the last two papers published a joint paper [3], In “Tetris is hard” they left open the question of the complexity of a variant of the game. In this variant, we can move and rotate a piece all the time we want in the first row, and the piece will not fall unless we move it down. Once we move the piece down we will not be able to move or rotate the piece anymore. This way to play tetris was introduced by John Brzustowsky [4]. In this paper we are going to prove that the problem of clearing the board in this variant of the game is NP-complete, solving a problem that has been open during thirteen years. To accomplish this task we are going to reduce the 3-partition problem to the problem of clearing a particular board on that variant of tetris.