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Encyclopedia > Maze

A maze is a complex tour puzzle in the form of a complex branching passage through which the solver must find a route. This is different from a labyrinth, which (strictly speaking) has an unambiguous through-route and is not designed to be difficult to navigate (despite the common uses of the word to indicate various complex, confusing structures). The pathways and walls in a maze or labyrinth are fixed (pre-determined). Maze-type puzzles where the given walls and paths may change during the game are covered under the main puzzle category of tour puzzles. A maze is a type of puzzle that consists of a complex branching passage through which the solver must find a route. ... Image File history File links Wikitext. ... In tour puzzles the player of the puzzle makes a trip around a (not necessarily two-dimensional) board using a token which represents a traveller. ... This article is about the mazelike structure from Greek mythology. ...

A small maze
A small maze

Contents

Wikipedia does not have an article with this exact name. ...

Maze construction

Mazes have been built with walls and rooms, with hedges, turf, or with paving stones of contrasting colors or designs, or in fields of crops such as corn or, indeed, maize. Maize mazes can be very large; they are usually only kept for one growing season, so they can be different every year, and are promoted as seasonal tourist attractions. One type of maze consists of a set of rooms linked by doors (so a passageway is just another room in this definition). Players enter at one spot, and exit at another, or the idea may be to reach a certain spot in the maze. Mazes can also be printed or drawn on paper to be followed by a pencil or fingertip. One of the short stories of Jorge Luis Borges featured a book, called The Garden of Forking Paths, that was a literary maze. Various maze generation algorithms exist for building mazes, either by hand or by computer. In gardening a hedge is a row of woody plants, generally of one species, used to demarcate spaces. ... Walking the turf maze at Wing, Rutland Historically, a turf maze is a labyrinth made by cutting a convoluted path into a level area of short grass, turf or lawn. ... This article is about cereals in general. ... This article is about the maize plant. ... This article does not cite any references or sources. ... For other uses, see Paper (disambiguation). ... This article is about the handwriting instrument. ... Borges redirects here. ... The Garden of Forking Paths (Spanish: El Jardín de senderos que se bifurcan) is a short story by Argentine writer and poet Jorge Luis Borges. ... There are a number of different maze generation algorithms, that is, automated methods for the creation of mazes. ... This article is about the machine. ...


Generating mazes

Computer generated maze

There are many different approaches to automate the generation of mazes. computer generated maze File links The following pages link to this file: Maze Categories: GFDL images | Creative Commons Attribution-ShareAlike 1. ... There are a number of different maze generation algorithms, that is, automated methods for the creation of mazes. ...


Stack-based approach

This is one of the simplest ways to generate a maze using a computer. Consider the space for a maze being a large grid of cells (like a large chess board), each cell starting with four walls. Starting from a random cell, the computer then selects a random neighbouring cell that has not yet been visited. The computer removes the 'wall' between the two cells and adds the new cell to a stack (this is analogous to drawing the line on the floor). The computer continues this process, with a cell that has no unvisited neighbours being considered a dead-end. When at a dead-end it backtracks through the path until it reaches a cell with an unvisited neighbour, continuing the path generation by visiting this new, unvisited cell (creating a new junction). This process continues until every cell has been visited, causing the computer to backtrack all the way back to the beginning cell. This approach guarantees that the maze space is completely visited.


As stated, the algorithm is very simple and does not produce overly-complex mazes. More specific refinements to the algorithm can help to generate mazes that are harder to solve.


Solving mazes

The mathematician Leonhard Euler was one of the first to analyse plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...


The following algorithms are designed to be used inside the maze by a traveler with no prior knowledge of the maze. Flowcharts are often used to represent algorithms. ...


Random mouse algorithm

This is a trivial method that can be implemented by a very unintelligent robot or perhaps a mouse. It is simply to proceed in a straight line until an obstruction is reached, and then to make a random decision about the next direction to follow. For other uses, see robot (disambiguation). ...


Wall follower

The wall follower, the best-known rule for traversing mazes, is also known as either the left-hand rule or the right-hand rule. If the maze is simply connected, that is, all its walls are connected together or to the maze's outer boundary, then by keeping one hand in contact with one wall of the maze the player is guaranteed not to get lost and will reach a different exit if there is one; otherwise, he or she will return to the entrance. If the maze is not simply connected (i.e. if the start or endpoints are in the center of the structure or the pathways cross over and under each other), this method will not be guaranteed to help the goal to be reached. Image File history File links BoBold textBold textItalic textBold textInsert non-formatted text here ld text0i[m0p9k[-9,[]-i == Headline text == == Headline text ==Media:Example. ... In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. ...


Wall-following can be done in 3D or higher dimensional mazes if its higher dimensional passages can be projected onto the 2D plane in a deterministic manner. For example, if in a 3D maze "up" passages can be assumed to lead northwest, and "down" passages can be assumed to lead southeast, then standard wall following rules can then be applied.


Pledge algorithm

Robot in a wooden maze
Robot in a wooden maze

Disjoint mazes can still be solved with the wall follower method, if the entrance and exit to the maze are on the outer walls of the maze. If however, the solver starts inside the maze, it might be on a section disjoint from the exit, and wall followers will continually go around their ring. The Pledge algorithm (named after Jon Pledge of Exeter) can solve this problem (see "Turtle Geometry: the computer as a medium for exploring mathematics", Abelson & diSessa, 1980). Image File history File linksMetadata Download high resolution version (2048x1536, 983 KB) Blind Cyclope in a maze at EPFL / Switzerland. ... Image File history File linksMetadata Download high resolution version (2048x1536, 983 KB) Blind Cyclope in a maze at EPFL / Switzerland. ... The city of Exeter is the county town of Devon, in the southwest of England, also known as the West Country. ...


The Pledge algorithm, designed to circumvent obstacles, requires an arbitrarily chosen direction to go toward. When an obstacle is met, one hand (say the right hand) is kept along the obstacle while the angles turned are counted. When the solver is facing the original direction again, and the angular sum of the turns made is 0, the solver leaves the obstacle and continues moving in its original direction.


Note that the use of "total turning" rather than just the "current direction" allows the algorithm to avoid traps shaped like an upper case "G". If one proceeds left into the trap, one gets turned around a full 360 degrees by the walls. A naive "current direction" algorithm gets into a limit cycle as it leaves the lower rightmost wall heading left and runs into the curved section on the left again. The Pledge algorithm does not leave the rightmost wall due to the total turning not being zero at that point. It follows the wall all the way around, finally leaving it heading left on the bottom outside.


This algorithm allows a person with a compass to find his way from any point inside to an outer exit of any finite and fair two-dimensional maze, regardless of the initial position of the solver. However, this algorithm will not work in doing the reverse, namely finding the way from an entrance on the outside of a maze to some end goal within it.


Tremaux's algorithm

Tremaux's algorithm is an efficient method to find the way out of a maze that requires drawing a line on the floor to mark a path, and is guaranteed to work for all mazes that have well-defined passages. On arriving at a junction, pick a direction and mark it and the direction you came from. When arriving at a marked junction pick an unmarked passage if possible. If it is not possible to pick an unmarked passage take a marked one, marking it again (you can pick the path you came from). Never pick a twice marked path, where you will never need to take any passage more than twice. If there is no exit, this method will take you back to the start. [1]


Tremaux's algorithm was referenced in The Simpsons episode Stop, or My Dog Will Shoot! when Lisa proposes this method for finding the exit of a tricky corn maze. Simpsons redirects here. ... Stop, or My Dog Will Shoot! is the 20th episode of The Simpsons eighteenth season, which originally aired May 13, 2007. ...


Mazes in science experiments

Mazes are often used in science experiments to study spatial navigation and learning. Such experiments typically use rats or mice. Examples are In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... Learning is the acquisition and development of memories and behaviors, including skills, knowledge, understanding, values, and wisdom. ... Species 50 species; see text *Several subfamilies of Muroids include animals called rats. ... This article is about the animal. ...

The Barnes maze is used to measure spatial learning and memory. ... In neuroscience, the Morris water maze is a behavioral procedure designed to test spatial memory. ... The Radial Arm Maze is used to measure Spatial learning and Memory. ...

Other types of mazes

Logic mazes
See Logic maze. These mazes use same rule as the other mazes other than "don't cross the lines" to restrict motion. Examples are
  • Area-mazes or A-mazes, which the area of the tile stepped on must alternately increase and decrease with every step.
  • Dice mazes, in which a die is rolled onto cells based on various rules.
  • Number Mazes, in which a grid of numbers is navigated by traveling the number shown on the current square.
  • Multi-State mazes, in which the rules for navigation change depending on how the maze has been navigated.
Mazes in higher dimensions
It is possible for a maze to have three or more dimensions. A maze with bridges is three dimensional, and some natural cave systems are three dimensional mazes. The computer game Descent utilized fully three dimensional mazes. Any maze can be topologically mapped onto a three-dimensional maze.
Picture maze
A maze that forms a picture when solved. See picture maze.
Dead end maze
A maze game where the route creates the dead ends.
Turf mazes and Mizmazes
A pattern like a long rope folded up, without any junctions or crossings.

Logic mazes, sometimes called mazes with rules, are logic puzzles with all the aspects of a tour puzzle that fall outside of the scope of a typical maze. ... Descent is a 3D first-person shooter video game which spawned two direct sequels (Descent II and Descent³). The Descent name was also used for an unrelated space simulator released by Volition, Inc. ... Airplane, 60x40 picture maze Picture mazes are maze puzzles that form pictures when solved the same as traditional mazes // Rules picture mazes require no special rules or learning and rules are very simple: - Draw a path from the entrance to exit of the puzzle, avoiding the dead ends - Fill the... Walking the turf maze at Wing, Rutland Historically, a turf maze is a labyrinth made by cutting a convoluted path into a level area of short grass, turf or lawn. ... Mizmaze (or Miz-Maze) is the name given to two of Englands eight surviving historic turf mazes. ...

Publications about mazes

Numerous mazes of different kinds have been drawn, painted, published in books and periodicals, used in advertising, in software, and sold as art. In the 1970s there occurred a publishing "maze craze" in which numerous books, and some magazines, were commercially available in nationwide outlets and devoted exclusively to mazes of a complexity that was able to challenge adults as well as children (for whom simple maze puzzles have long been provided both before, during, and since the 1970s "craze").

Inside the labyrinth of villa Pisani at Stra, near Venice
Inside the labyrinth of villa Pisani at Stra, near Venice

Some of the best-selling books in the 1970s and early 1980s included those produced by Vladimir Koziakin, Rick and Glory Brightfield, Dave Phillips, Larry Evans, and Greg Bright. Koziakin's works were predominantly of the standard two-dimensional "trace a line between the walls" variety. The works of the Brightfields had a similar two-dimensional form but used a variety of graphics-oriented "path obscuring" techniques - although the routing was comparable to or simpler than Koziakin's mazes, the Brightfield's mazes did not allow the various pathway options to be discerned so easily by the roving eye as it glanced about. Greg Bright's works went beyond the standard published forms of the time by including "weave" mazes in which illustrated pathways can cross over and under each other. Bright's works also offered examples of extremely complex patterns of routing and optical illusions for the solver to work through. What Bright termed "mutually accessible centers" (The Great Maze Book, 1973) also called "braid" mazes, allowed a proliferation of paths flowing in spiral patterns from a central nexus and, rather than relying on "dead ends" to hinder progress, instead relied on an overabundance of pathway choices. Rather than have a single solution to the maze, Bright's routing often offered multiple equally valid routes from start to finish, with no loss of complexity or diminishment of solver difficulties because the result was that it became difficult for a solver to definitively "rule out" a particular pathway as unproductive. Some of Bright's innovative mazes had no "dead ends" - although some clearly had looping sections (or "islands") that would cause careless explorers to keep looping back again and again to pathways they had already travelled. The books of Larry Evans focused on 3-D structures, often with realistic perspective and architectural themes, and Bernard Meyers (Supermazes No. 1) produced similar illustrations. Both Greg Bright (The Hole Maze Book) and Dave Phillips (The World's Most Difficult Maze) published maze books in which the sides of pages could be crossed over and in which holes could allow the pathways to cross from one page to another, and one side of a page to the other, thus enhancing the 3-D routing capacity of 2-D printed illustrations. Image File history File linksMetadata Size of this preview: 450 × 600 pixelsFull resolution (1200 × 1600 pixel, file size: 1. ... Image File history File linksMetadata Size of this preview: 450 × 600 pixelsFull resolution (1200 × 1600 pixel, file size: 1. ... Villa Pisani This villa was likely built around 1555 as a result of Alvise Pisanis appointment to doge. ... Stra is a town in the province of Venice, Veneto, Italy. ... For other uses, see Venice (disambiguation). ...

Maze at St. Louis Botantical Gardens
Maze at St. Louis Botantical Gardens

A recent book by Galen Wadzinski (The Ultimate Maze Book) offers formalized rules for more recent innovations that involve single-directional pathways, 3-D simulating illustrations, "key" and "ordered stop" mazes in which items must be collected or visited in particular orders to add to the difficulties of routing (such restrictions on pathway traveling and re-use are important in a printed book in which the limited amount of space on a printed page would otherwise place clear limits on the amount of choices and pathways that can be contained within a single maze). Although these innovations are not all entirely new with Wadzinski, the book marks a significant advancement in published maze puzzles, offering expansions on the traditional puzzles that seem to have been fully informed by various video game innovations and designs, and adds new levels of challenge and complexity in both the design and the goals offered to the puzzle-solver in a printed format. Image File history File linksMetadata Size of this preview: 800 × 600 pixelsFull resolution (1280 × 960 pixel, file size: 303 KB, MIME type: image/jpeg) the maze at the St. ... Image File history File linksMetadata Size of this preview: 800 × 600 pixelsFull resolution (1280 × 960 pixel, file size: 303 KB, MIME type: image/jpeg) the maze at the St. ...


Further reading

  • Adrian Fisher & Georg Gerster, The Art of the Maze, Weidenfeld & Nicolson (1990) ISBN 0-297-83027-9
  • Jeff Saward, Magical Paths, Mitchell Beazley (2002) ISBN 1-84000-573-4
  • W.H. Matthews, Mazes and Labyrinths: Their History and Development (1927). Includes Bibliography. Dover Publications (1970) ISBN 0-486-22614-X
  • H. Abelson and A. diSessa, Turtle Geometry: The Computer as a Medium for Exploring Mathematics (The MIT Press 1980).

Mazes open to the public

Public hedge maze in the "English Garden" at Schönbusch Park, Aschaffenburg, Germany
Public hedge maze in the "English Garden" at Schönbusch Park, Aschaffenburg, Germany

Image File history File linksMetadata Schönbusch4. ... Image File history File linksMetadata Schönbusch4. ... Aschaffenburg (IPA: ; dialect: [ˈaʒəˌbɜːʃ]) is a large town in north west Bavaria, Germany. ...

Europe

Altjeßnitz is a municipality in the district of Anhalt-Bitterfeld, in Saxony-Anhalt, Germany. ... Statistics Province: Ulster County: District: Down District UK Parliament: South Down European Parliament: Northern Ireland Dialling Code: 028, +437 Post Town: Castlewellan Postal District(s): BT31 Area:  ? km² Population (2001) 2,392 Castlewellan Village Castlewellan Lake and Forest Park Castlewellan (in Irish: Caisleán Uidhilín, ie Uidhilín’s... A view of Chatsworth from the south-west circa 1880. ... Hampton Court redirects here. ... Kentwell Hall is a stately home in Suffolk, England. ... The church of the Holy Trinity Long Melford (or Melford, as it is more generally known) is a large, ancient village in the county of Suffolk, England, on the border with Essex, which is marked by the River Stour, approximately 20 miles from Colchester and 20 miles from Bury St. ... Suffolk (pronounced ) is a large historic and modern non-metropolitan county in East Anglia, England. ... The front of Leeds Castle Leeds Castle Leeds Castle in Winter Leeds Castle and its ditch Leeds Castle, four miles south east of Maidstone, Kent, England, dates back to 1119, though a manor house stood on the same site from the 9th century. ... Adrian Fisher is internationally recognised as a leading maze designer. ... A drawing of Longleat in the early 18th century by Leonard Knyff. ... Moomin World or Moomin Theme Park (especially for kids) is situated in Naantali near the city of Turku in Western Finland. ... Scone Palace. ... Adrian Fisher is internationally recognised as a leading maze designer. ... Located 5 minutes from the M5 junction 19, Noahs Ark Zoo Farm is a recently developed tourist attraction in Wraxall, North Somerset. ... Paultons Park is an amusement park situated in the village of Ower near Romsey in Hampshire, UK. The name is derived from Paultons House, the house that used to be on the grounds before a fire. ... Saffron Walden is a small market town in the Uttlesford district of Essex, England. ... Samsø is an island in the North Sea bay of Kattegat 15 kilometers off the Jutland Peninsula. ... Schönbrunn Palace (Schloss Schönbrunn) in Vienna is one of the most important cultural monuments in Austria and since the 1860s has also been one of the major tourist attractions in Vienna. ... Symonds Yat is a village within the Forest of Dean and a popular tourist destination straddling the River Wye on the county border of Herefordshire and Gloucestershire in England. ... For other uses, see Crystal Palace. ... A copse is an English term for a small lowland woodland. ... Remains of the old hall, Worden Park Worden Park is a large area of parkland situated on the outskirts of Leyland, a town in the borough of South Ribble, Lancashire, United Kingdom. ... Leyland is a town in the borough of South Ribble, Lancashire, United Kingdom, approximately 6 miles south of Preston. ...

North America

There are very few or no other articles that link to this one. ... City of Shakopee Logo Shakopee is a city in Scott County, Minnesota, United States. ... Capital Saint Paul Largest city Minneapolis Area  Ranked 12th  - Total 87,014 sq mi (225,365 km²)  - Width 250 miles (400 km)  - Length 400 miles (645 km)  - % water 8. ... Garden City is a census-designated place (CDP) in Horry County, South Carolina, United States. ... Magnolia Plantation, located 13 miles (21 km) north of Charleston, South Carolina was founded in 1676 on the Ashley River and is one of the oldest plantations in the south. ... Official language(s) English Capital Columbia Largest city Columbia Largest metro area Columbia Area  Ranked 40th  - Total 34,726 sq mi (82,965 km²)  - Width 200 miles (320 km)  - Length 260 miles (420 km)  - % water 6  - Latitude 32° 2′ N to 35° 13′ N  - Longitude 78° 32′ W to 83... For other uses, see Hay (disambiguation). ... Ashland is a town located in Hanover County, Virginia. ... Sterling is a town located in Worcester County, Massachusetts, USA. As of the 2000 census, the town had a total population of 7,257. ... This article needs cleanup. ... Template:Hide = Motto: Template:Unhide = Diversity Our Strength Image:Toronto, Ontario Location. ... James Dole, founder of the Hawaiian pineapple industry, is immortalized as the Pineapple King. ... Wahiawā is a census-designated place and town located more or less in the center of Oahu Island, on the plateau or central valley between the two volcanic mountains that comprise the island. ... Official language(s) English, Hawaiian Capital Honolulu Largest city Honolulu Area  Ranked 43rd  - Total 10,931 sq mi (29,311 km²)  - Width n/a miles (n/a km)  - Length 1,522 miles (2,450 km)  - % water 41. ... Mohonk Mountain House, taken from Skytop The Mohonk Mountain house is a large resort located atop the Shawangunk Ridge on the shore of Lake Mohonk, a half mile (800 m) long, 60 foot (20 m) deep lake. ... New Paltz is both a village and town in the U.S. state of New York. ...

Africa

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