definition of Wikipedia
A tetromino is a geometric shape composed of four squares, connected orthogonally. This, like dominoes and pentominoes, is a particular type of polyomino. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally.
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Polyominos are formed by joining unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other.
A free tetromino is a free polyomino made from four squares. There are five free tetrominos (see figure).
One-sided tetrominos are tetrominos that may be translated and rotated but not reflected. They are used by, and are overwhelmingly associated with, the game Tetris. There are seven distinct one-sided tetrominos. Of these seven, three have reflectional symmetry, so it does not matter whether they are considered as free tetrominos or one-sided tetrominos. These tetrominos are:
The remaining four tetrominos exhibit a phenomenon called chirality. These four come in two sets of two. Each of the members of these sets is the reflection of the other:
As free tetrominos, J is equivalent to L and S is equivalent to Z. But in two dimensions and without reflections, it is not possible to transform J into L or S into Z.
The fixed tetrominos allow only translation, not rotation or reflection. There are two distinct fixed I-tetrominos, four J, four L, one O, two S, four T, and two Z, for a total of 19 fixed tetrominos.
Although a complete set of free tetrominos has a total of 20 squares, and a complete set of one-sided tetrominos has 28 squares, it is not possible to pack them into a rectangle, like hexominoes and unlike pentominoes. The proof is that a rectangle covered with a checkerboard pattern will have 10 or 14 each of light and dark squares, while a complete set of free tetrominos (pictured) has 11 light squares and 9 dark squares, and a complete set of one-sided tetrominos has 15 light squares and 13 dark squares.
A bag including two of each free tetromino, which has a total area of 40 squares, can fit in 4×10 and 5×8 cell rectangles. Likewise, two sets of one-sided tetrominos can be fit to a rectangle in more than one way. The corresponding tetracubes can also fit in 2×4×5 and 2×2×10 boxes.
layer 1 : layer 2 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
layer 1 : layer 2 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
Each of the five free tetrominos has a corresponding tetracube, which is the tetromino extruded by one unit. J and L are the same tetracube, as are S and Z, because one may be rotated around an axis parallel to the tetromino's plane to form the other. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube:
In 3D, these eight tetracubes (suppose each piece consists of four 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:
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
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, the remaining seven pieces can fill a 7×2×2 box. (C represents D or S.)
layer 1 : layer 2 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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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
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