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Tour Puzzle - Game Web |
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 In tour puzzles the player of the puzzle makes a trip around a board (usually but not necessarily two-dimensional) using a token which represents a traveller.
Sometimes (as in mazes) the player him/herself makes the trip. Sometimes the player has more than one token with which to travel. Sometimes certain objects have to be found and/or retrieved on the way.
Often there is a given start and finish position for the player's token. Some tour puzzles demand that certain points on the board have to be visited on the way. Maze puzzles are often of this type.
Examples of tour puzzles
Knight's Tour
The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once.
There are a great many solutions to the problem, of which exactly 26,534,728,821,064 have the knight finishing on a square from which it attacks the starting square, on an 8x8 board. Such a tour is described as directed and closed. (A directed tour is a directed graph: the direction of the tour is specified. A closed tour is one that ends on the starting square.) The number of undirected closed tours is half this number, since every tour can be traced in reverse. Otherwise the tour is open (as in the first diagram). There are 9,862 undirected closed tours on a 6x6 board, and no such tours on smaller boards.
Many variations on this topic have been studied by mathematicians, including Euler, over the centuries using:- Differently sized boards.
- Two-player games based on this idea.
- Problems using slight variations on the way the knight moves.
The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory, which is NP-complete. The problem of getting a closed knight's tour is similarly an instance of the hamiltonian cycle problem. Note however that, unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time.
The pattern of a Knight's Tour on a half-board has been presented in verse form (as a literary constraint) in the highly stylized Sanskrit poem Kavyalankara written by the 9th century Kashmiri poet Rudrata, which discusses the art of poetry, especially with relation to theater (Natyashastra). As was often the practice in ornate Sanskrit poetry, the syllabic patterns of this poem elucidate a completely different motif, in this case an open knight's tour on a half-chessboard.
The first algorithm for completing the Knight's Tour was Warnsdorff's algorithm, first described in 1823 by H. C. Warnsdorff.
In the 20th century the Oulipo group of writers used it among many others. The most notable example is the 10 × 10 Knight's Tour which sets the order of the chapters in Georges Perec's novel Life: A User's Manual.
Mazes - 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 has an actual 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.
Labyrinth
In Greek mythology, the Labyrinth (labyrinthos) was an elaborate structure constructed for King Minos of Crete at Knossos and designed by the legendary artificer Daedalus to hold the Minotaur, a creature that was half man and half bull and which was eventually killed by the Athenian hero Theseus. Daedalus had made the Labyrinth so cunningly that he himself could barely escape it after he built it. Theseus was aided by Ariadne, who provided him with a fateful thread, literally the "clew," or "clue," to wind his way back again.
The term labyrinth is often used interchangeably with maze, but modern scholars of the subject use a stricter definition. For them, a maze is a tour puzzle in the form of a complex branching passage with choices of path and direction; while a single-path ("unicursal") labyrinth has only a single Eulerian path to the center. A labyrinth has an unambiguous through-route to the center and back and is not designed to be difficult to navigate.
This unicursal design was wide-spread in artistic depictions of the Minotaur's Labyrinth even though both logic and literary descriptions of it make it clear that the Minotaur was trapped in a multicursal maze.
A labyrinth can be represented both symbolically and/or physically. Symbolically it is represented in art or designs on pottery, as body art, etched on walls of caves, etc. Physical representations are common throughout the world, and are generally constructed on the ground so they may be walked along from entry point to center and back again. They have historically been used in both group ritual and for private meditation.
Mizmazes
Mizmaze (or Miz-Maze) is the name given to two of England's eight surviving historic turf mazes. One is at Breamore, in Hampshire; the other is on top of St Catherine's Hill, overlooking the city of Winchester, Hampshire.
A mizmaze forms a pattern unlike conventional mazes and is classed as a labyrinth because the path has no junctions or crossings. The pattern appears more like a very long rope, neatly arranged to fill the area.
Logic mazes
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. These mazes have special rules, sometimes including multiple states of the maze or navigator. Popular logic mazes include tilt mazes and other novel designs which usually increase the complexity of the maze, sometimes to the point that the maze has to be designed by a program to eliminate multiple paths. Additional examples include:- Area-mazes or A-mazes, which the area of the tile stepped on must alternately increase and decrease with every step.
- Rolling 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.
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