Editing Tetromino (section)

==Tiling a rectangle==
===Filling a rectangle with one set of tetrominoes===
A single set of free tetrominoes or one-sided tetrominoes cannot fit in a rectangle. This can be shown with a proof similar to the [[Mutilated chessboard problem|mutilated chessboard argument]]. A 5×4 rectangle with a checkerboard pattern has 20 squares, containing 10 light squares and 10 dark squares, but a complete set of free tetrominoes has either 11 dark squares and 9 light squares, or 11 light squares and 9 dark squares. This is due to the T tetromino having either 3 dark squares and one light square, or 3 light squares and one dark square, while all other tetrominoes each have 2 dark squares and 2 light squares. Similarly, a 7×4 rectangle has 28 squares, containing 14 squares of each shade, but the set of one-sided tetrominoes has either 15 dark squares and 13 light squares, or 15 light squares and 13 dark squares. By extension, any odd number of sets for either type cannot fit in a rectangle. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. This was discovered by exhausting all possibilities in a computer search.

{{multiple image 
| align = left
| image_style = border:none;
| image_gap = 25
| image1 = Tetrominoes with Checkerboard Squares.svg
| width1=690
| caption1 = The free tetrominoes (left side of line) have 11 dark squares and 9 light squares.<br>The one-sided tetrominoes (all 7 shown above) have 15 dark squares and 13 light squares.
| image2 = Checkerboard Rectangle 7x4.svg
| width2=240
| caption2 = A 5×4 board has 10 squares each color.<br>A 7×4 board has 14 squares each color.
}}
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=== Parity ===
A further consequence of the T tetromino having 3 squares of one colour and 1 square of the other is that any rectangle containing an even number of squares must contain an even number of T tetrominoes. Conversely, any rectangles containing an odd number of squares must contain an odd number of T tetrominoes.

===Filling a modified rectangle with one set of tetrominoes===
All three sets of tetrominoes can fit rectangles with holes: 
<ul>
<li>All 5 free tetrominoes fit a 7×3 rectangle with a hole.</li>
<li>All 7 one-sided tetrominoes fit a 6×5 rectangle with two holes of the same "checkerboard color".</li>
<li>All 19 fixed tetrominoes fit a 11×7 rectangle with a hole.</li>
</ul>

{{multiple image 
| align = left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino Tiling - 7x3.svg
| width1 = 116
| caption1 = Free tetrominoes in a rectangle with one hole
| image2 = Tetromino Tiling - 6x5.svg
| width2 = 100
| caption2 = One-sided tetrominoes in a rectangle with two holes
| image3 = Tetromino Tiling 11x7.svg
| width3 = 175
| caption3 = Fixed tetrominoes in rectangle with one hole
}}
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===Filling a rectangle with two sets of tetrominoes===
Two sets of free or one-sided tetrominoes can fit into a rectangle in different ways, as shown below:

{{multiple image 
| align=left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino Tiling 8x5.svg
| width1=114
| caption1 = Two sets of free tetrominoes in a 5×8 rectangle
| image2 = Tetromino Tiling 10x4.svg
| width2=150
| caption2 = Two sets of free tetrominoes in a 4×10 rectangle
| image3 = Tetromino Tiling 8x7.svg
| width3=120
| caption3 = Two sets of one-sided tetrominoes in a 8×7 rectangle
| image4 = Tetromino Tiling 14x4.svg
| width4=200
| caption4 = Two sets of one-sided tetrominoes in a 14×4 rectangle
}}
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