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Tiling Puzzle - Game Web |
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 Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles.
Tiling puzzles may be made from wood, metal, cardboard, plastic or any other sheet-material. Many tiling puzzles are now available as computer games.
Tiling puzzles have a long history. Some of the oldest and most famous are jigsaw puzzles and the Tangram puzzle.
Other examples of tiling puzzles include:
Conway puzzle - Conway's puzzle is a packing problem using rectangular blocks, named after its inventor, mathematician John Conway. It calls for packing thirteen 1 × 2 × 4 blocks, one 2 × 2 × 2 block, one 1 × 2 × 2 block, and three 1 × 1 × 3 blocks into a 5 × 5 × 5 box.
Domino tiling, of which the mutilated chessboard problem is one example - A domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.
Eternity puzzle - The eternity puzzle was a geometric puzzle with a million-pound prize, created by Christopher Monckton, who put up half the money himself, the other half being put up by underwriters in the London insurance market. The puzzle was distributed by the Ertl Company.
Puzz-3D - Puzz-3D is the brand name of three-dimensional jigsaw puzzles, manufactured by Wrebbit, Inc. Unlike traditional puzzles which are composed of series of flat pieces with parts of an image on them, when put together, create a single unified image, the Puzz-3d series of puzzles are composed on plastic foam, with the part of an image graphed on a stiff paper facade glued to the underlying foam piece and cut to match the piece's dimensions. When the pieces are put together, they create a structure such as a building.
Squaring the square - A square with sides equal to a unit length multiplied by an integer is called an integral square. Squaring the square is the problem of tiling one integral square using only other integral squares. Squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square, where all contained squares are of different size.
Tantrix - Tantrix is a hexagonal tile-based abstract strategy game invented by Mike McManaway from New Zealand. Each of the 56 different Bakelite tiles in the set contains three lines, going from one edge of the tile to another. No two lines on a tile have the same colour. There are four colours in the set: red, yellow, blue, and green. No two tiles are identical, and each is individually numbered from 1 through 56.
Pentominoes - A pentomino is a polyomino composed of five (Greek πέντε / pente) congruent squares, connected orthogonally. There are twelve different pentominoes, and they are named after the letters of the Latin alphabet that they resemble. Ordinarily, the reflection symmetry and rotation symmetry of a pentomino does not count as a different pentomino.
Tangram
Tangram (literally "seven boards of skill") is a dissection puzzle. It consists of seven pieces, called tans, which fit together to form a shape of some sort. The objective is to form a specific shape with seven pieces. The shape has to contain all the pieces, which may not overlap.
Jigsaw puzzle
A jigsaw puzzle is a tiling puzzle that requires the assembly of numerous small, often oddly shaped, interlocking and tessellating pieces. Each piece has a small part of a picture on it; when complete, a jigsaw puzzle produces a complete picture.
Jigsaw puzzles were originally created by painting a picture on a flat, rectangular piece of wood, and then cutting that picture into small pieces with a jigsaw, hence the name. John Spilsbury, a London mapmaker and engraver, is credited with commercialising jigsaw puzzles around 1760.
Most modern jigsaw puzzles are made out of cardboard, since they are easier and cheaper to mass produce. An enlarged photograph or printed reproduction of a painting or other two-dimensional artwork is glued onto the cardboard before cutting. This board is then fed into a press. The press forces a set of hardened steel blades of the desired shape through the board until it is fully cut. This procedure is similar to making shaped cookies with a cookie cutter. The forces involved, however, are tremendously greater and a typical 1000-piece puzzle will require a press which can generate upwards of 700 tons of force to push the knives of the puzzle die through the board. A puzzle die comprises a flat board, often made from plywood, which has slots cut or burned in the same shape as the knives that will be used. These knives are set into the slots and covered in a compressible material, typically foam rubber, the function of which is the ejection of the cut puzzle pieces.
Typical images found on jigsaw puzzles include scenes from nature, buildings, and repetitive designs. Castles and mountains are two traditional subjects. However, any kind of picture can be used to make a jigsaw puzzle; some companies offer to turn personal photographs into puzzles. Completed puzzles can also be attached to a backing with adhesive to be used as artwork.
During recent years a range of jigsaw puzzle accessories including boards, cases, frames and roll-up mats has become available that are designed to assist jigsaw puzzle enthusiasts.
Mutilated chessboard problem
The mutilated chessboard problem is a tiling puzzle introduced by Martin Gardner in his Scientific American column "Mathematical Games." The problem is as follows:
Suppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
Most considerations of this problem in literature provide solutions "in the conceptual sense" without proofs. John McCarthy proposed it as a hard problem for automated proof systems. In fact, its resolution is exponentially hard.
Packing problem
Packing problems are one area where mathematics meets puzzles (recreational mathematics). Many of these problems stem from real-life packing problems.
In a packing problem, you are given:- One or more (usually two- or three-dimensional) containers.
- Several 'goods', some or all of which must be packed into this container.
Usually the packing must be without gaps or overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimised. In others, gaps are allowed, but overlaps are not (usually the total area of gaps has to be minimised).
Dissection puzzle
A dissection puzzle, also called a transformation puzzle is a tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of dissection puzzle. Puzzles may include various restraints, such as hinged pieces, pieces that can fold, or pieces that can twist. Creators of new dissection puzzles emphasize using a minimum number of pieces, or creating novel situations, such as ensuring that every piece connects to another with a hinge.
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