Square Tetris
The New Tetris, Square mode of Tetris Worlds, and Lockjaw: The Overdose reward the player for creating 4x4 block squares out of four tetrominoes. Building wide platforms (flat top sides) into the top of a square in progress allows more than one such square to be built at once, keeping options open for handling any tetromino.
Monosquares
It is best to use rotational symmetry (also called the "spiral" method or the "swastika" method) to make monosquares with platforms 3 blocks wide.
Spiral L monosquare, on both sides:
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Spiral J monosquare, on both sides:
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Spiral T monosquare, on both sides:
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L on the left and J on the right require a double slide (see below).
Multisquares
SJSJ | ZLZL:
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JSSJ | LZZL:
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ZLTT | SJTT:
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ZLLZ | SJJS (This is possible in The New Tetris but relies on uncommon wall kicks. The trick is to have the Z or S in the right rotation and take it down so that the bottom square is resting on the top square of the outer L or J, then rotate away and then towards the construction in quick succession.):
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STTJ | ZTTL:
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TZTJ | TSTL:
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LTZT | JTST:
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LTTI | JTTI:
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LSIL | JZIJ (requires double slide):
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Double slide
The New Tetris uses smooth falling animation for tetrominoes. The way this implementation mixes the falling animation with collision detection means that a tetromino can normally slide only one block under an overhang unless there is something on the floor to support it.
There are, however, ways of getting a new piece further under the overhang in The New Tetris by rotating the piece in the right direction at the right time.