In the context of Tetris, fan games are computer tetromino games unconnected with Elorg, The Tetris Company, or any other licensor of rights in the Tetris trademark. They tend to follow the well-known rules of Tetris but may not strictly follow any official guidelines.
The following games more or less faithfully replicate the rules of some version of Tetris, either pre-Guideline or post-Guideline:
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4-Tris - Recent game cartridge by Joe Zbiciak for the old Mattel Intellivision game console
Aint - free software for MS-DOS, possibly the first copylefted tetromino game
Bastet - Literally, Bastard Tetris. Made to be as annoying and frustrating as possible.
BCTMP - Brian Caos Tempting Music Player (versions Beta 1 and Beta 2 previously known as "Brian Caos Tetris Music Player"), IT/MOD/S3M/XM music player with a built-in Tetris game
Bedter by Andrew Bednarz - freeware for Windows and Linux, famously targeted by TTC which alleged "look and feel" copyright violation
Block Star - Integrated Facebook Tetris, featuring comparison leaderboard to your circle of friends, or all users.
Breakris by Jürgen Müller - freeware for Windows. It has a very good single and multiplayer-Tetris, plus Breakris - Breakout and Tetris at the same time.
Java Tetris v2.11 - Nice Java applet Tetris clone by Andreas Ludwig that supports multi-player and computer AI players
Lockjaw - primarily a guideline simulator, with dozens of options to change behavior
Lockjaw: The Overdose (formerly Tetanus On Drugs) - free software for Game Boy Advance, inspired by Tetripz and considered by many players to be superior in look and feel to the GBA version of Tetris Worlds
Quinn - Described as an arcade-style tetromino puzzle game written and designed specifically for Mac OS X
These games have game play rules inspired by Tetris, such as rotation of shapes to form complete lines, but introduce rule variations that change the game play greatly.
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3DTRIS - 3dtris.de A clone of Blockout (3D variant of Tetris) in Macromedia Flash. Its gameplay is very smooth, just like original Blockout. Online high score list with growing competitions.
Blockout II - blockout.net The best clone of Blockout (3D Tetris) for Windows using DirectX. Online high score list, forum, wiki.
Quadra - Open source clone with cascade gravity. Features very fast-paced online play.
Java Tetris - A Java Tetris game that follows the gameplay rules of the original, but has more than 30 shape types and a few new game modes, including a "Hardcore" mode where the game is played on a 125x125 grid.
3D Tetris - 3dtetris.com 3D variant of the classic Tetris game. You can play the game: online, offline and mobile. The game use the Java virtual machine.