Tetromino

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A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally. This is a particular type of polyomino, like dominoes and pentominoes are. Sometimes the term is generalized to apply to configurations of four orthogonally connected cubes.

A popular use of tetrominoes is in the video game Tetris. However, the Tetris spelling of the word differs slightly (for obvious reason) by replacing the first 'o' with an 'i' to make the word Tetrimino.

Counting rotations in two dimensions as equivalent, there are seven possible shapes:

  • Image:Tetris-i.png I (also called "stick", "straight", "long"): four blocks in a straight line
  • Image:Tetris-o.png O (also called "square", "package", "block"): four blocks in a 2×2 square
  • Image:Tetris-t.png T: a row of three blocks with one added below the center. A common move with the T piece is to spin it in place to fill a line.
  • Image:Tetris-l.png L: a row of three blocks with one added below the left side
  • Image:Tetris-j.png J (also called "inverted L" or "Gamma"): a row of three blocks with one added below the right side. This piece is a reflection of L but cannot be rotated into L in two dimensions; this is an example of chirality. However, in three dimensions, this piece is identical to L.
  • Image:Tetris-s.png S: bent trimino with block placed on outside of clockwise side
  • Image:Tetris-z.png Z: bent trimino with block added on outside of anticlockwise side. The same symmetry properties as with L and J apply with S and Z.

Adding a third dimension produces three more patterns, all created by placing a unit cube on the bent trimino:

However, going to three dimensions means that rotation is allowed in three dimensions. Thus, the two L-shaped and S-shaped pieces are now equivalent.

Some people refer to the pieces by the colour in which they are drawn in a particular implementation of the Tetris game, but those colours vary from implementation to implementation so this is not very sensible. For example, in many older versions of Tetris, the red piece is I.

Colors of pieces in various implementations
Piece Vadim Gerasimov's
original Tetris 		Microsoft Tetris Tetris: The Grand Master The New Tetris Tetris DS
I red red red cyan cyan
O blue cyan yellow white yellow
T brown gray cyan yellow magenta
L magenta yellow orange magenta orange
J white magenta blue blue blue
S green blue magenta green green
Z cyan green green red red

Note that the box art of The New Tetris reverses the colors of the L and J pieces and the S and Z pieces.

Contents

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Filling the box with 3D pieces

In 3D, these 8 pieces (suppose each piece consists of 4 cubes, L and J are the same. Z and S are the same) can fit in a 4×4×2 or 8×2×2 box. The following is one of the solutions. D, S and B represent right screw, left screw and branch point, respectively

4×4×2 box

layer 1  :  layer 2

S T T T  :  S Z Z B
S S T B  :  Z Z B B
O O L D  :  L L L D
O O D D  :  I I I I

8×2×2 box 

    layer 1     :     layer 2

D Z Z L O T T T : D L L L O B S S
D D Z Z O B T S : I I I I O B B S

If chiral pairs (D and S) are considered as identical, remaining 7 pieces can fill 7×2×2 box. (C represents D or S.) 

L L L Z Z B B : L C O O Z Z B
C I I I I T B : C C O O T T T
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Tiling the rectangle and filling the box with 2D pieces

Although a complete set of 2D tetrominoes has a total of 20 (or 28, when mirror images count) squares, it is not possible to pack them into a rectangle, unlike pentominoes. A parity argument. which is essentially same as hexomino, can prove this.

Two sets of pieces (two of each piece), which have total area of 40, can fit in 4×10 and 5×8 rectangles. These can also fit in 2×4×5 and 2×2×10 box with 3D configuration.

5×8 rectangle 

L Z I I I I O O
L Z Z T T T O O
L L Z t T l l l
o o t t t z z l
o o i i i i z z

4×10 rectangle 

L L L Z Z I I I I i
L T T T Z Z l l l i
o o T z z t l O O i
o o z z t t t O O i

2×4×5 box 

Z Z T t I   l T T T i
L Z Z t I   l l l t i
L z z t I   o o z z i
L L O O I   o o O O i

2×2×10 box 

L L L z z Z Z T O O       o o z z Z Z T T T l
L I I I I t t t O O       o o i i i i t l l l

As a puzzle, these are relatively easy.

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See also

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External links

pt:Tetraminó

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