1.
Puzzle
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A puzzle is a game, problem, or toy that tests a persons ingenuity or knowledge. In a puzzle, one is required to put together in a logical way. There are different types of puzzles for different ages, such as puzzles, word-search puzzles, number puzzles. Puzzles are often devised as a form of entertainment but they can arise from serious mathematical or logistical problems. In such cases, their solution may be a significant contribution to mathematical research, solutions of puzzles often require the recognition of patterns and the creation of a particular kind of order. Sometimes not because of how complicated and diagonal the pattern can get, people with a high level of inductive reasoning aptitude may be better at solving such puzzles than others. But puzzles based upon inquiry and discovery may be solved easily by those with good deduction skills. Some notable creators of puzzles are Sam Loyd, Henry Dudeney, Boris Kordemsky and, more recently, David J. Bodycombe, Will Shortz, Lloyd King, the 1989 edition of the Oxford English Dictionary dates the word puzzle to the end of the 16th century. Its first documented use was in a book titled The Voyage of Robert Dudley. to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, the word later came to be used as a noun. The word puzzle comes from pusle, meaning bewilder, confound, the use of the word to mean a toy contrived to test ones ingenuity is relatively recent. He then used the pieces as an aid to the teaching of geography. After becoming popular among the public, this kind of teaching aid remained the primary use of jigsaw puzzles until about 1820, by the early 20th century, magazines and newspapers had found that they could increase their readership by publishing puzzle contests. The largest puzzle is made by German game company Ravensburger, the smallest puzzle ever made was created at LaserZentrum Hannover. It is only five square millimetres, the size of a dust grain, there are organizations and events that cater to puzzle enthusiasts, such as the World Puzzle Championship, the National Puzzlers League, and Ravenchase. There are also puzzlehunts, such as the Maze of Games, Puzzles can be divided into categories. For example, a maze is a type of tour puzzle, some other categories are construction puzzles, stick puzzles, tiling puzzles, transport puzzles, disentanglement puzzles, lock puzzles, folding puzzles, combination puzzles, and mechanical puzzles. A chess problem is a puzzle that uses chess pieces on a chess board, examples are the knights tour and the eight queens puzzle. Peg solitaire A puzzle box is a puzzle that can be used to hide something — jewelry, rubiks Cube and other combination puzzles can be stimulating toys for children or recreational activities for adults
2.
Guessing game
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A guess is a swift conclusion drawn from data directly at hand, and held as probable or tentative, while the person making the guess admittedly lacks material for a greater degree of certainty. In many of its uses, the meaning of guessing is assumed as implicitly understood, guessing may combine elements of deduction, induction, abduction, and the purely random selection of one choice from a set of given options. Guessing may also involve the intuition of the guesser, who may have a gut feeling about which answer is correct without necessarily being able to articulate a reason for having this feeling. Tschaepe quotes the description given by William Whewell, who says that this goes on so rapidly that we cannot trace it in its successive steps. A guess that is merely a hunch or is groundless. is arbitrary, a guess made with no factual basis for its correctness may be called a wild guess. Jonathan Baron has said that he value of a guess is l/N + l/N - l/N = l/N. Philosopher David Stove described this process as follows, A paradigm case of guessing is, when captains toss a coin to start a cricket match and this cannot be a case of knowledge, scientific knowledge or any other, if it is a case of guessing. If the captain knows that the coin will fall heads, it is just logically impossible for him also to guess that it will. More than that, however, guessing, at least in such a paradigm case, does not even belong on what may be called the epistemic scale. That is, if the captain, when he heads, is guessing, he is not, in virtue of that, believing, or inclining to think, or conjecturing, or anything of that sort. And in fact, of course, he normally is not doing any of these things when he guesses, and this is guessing, whatever else is. In such an instance, there not only is no reason for favoring heads or tails, Tschaepe also addresses the guess made in a coin flip, contending that it merely represents an extremely limited case of guessing a random number. Allows for combining abductive reasoning with deductive and inductive reasoning, in Jane Austens Emma, however, the author has the character, Emma, respond to a character calling a match that she made a lucky guess by saying that a lucky guess is never merely luck. There is always some talent in it, by contrast, a guess made using prior knowledge to eliminate clearly wrong possibilities may be called an informed guess or an educated guess. Uninformed guesses can be distinguished from the kind of informed guesses that lead to the development of a scientific hypothesis, Tschaepe notes that his process of guessing is distinct from that of a coin toss or picking a number. It has also noted that hen a decision must be made. A guess, however, may also be purely a matter of selecting one possible answer from the set of possible answers, Tschaepe notes that guessing has been indicated as an important part of scientific processes, especially with regard to hypothesis-generation. Regarding scientific hypothesis-generation, Tschaepe has stated that guessing is the initial, following the work of Charles S. Peirce, guessing is a combination of musing and logical analysis
3.
Riddle
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A riddle is a statement or question or phrase having a double or veiled meaning, put forth as a puzzle to be solved. Defining riddles precisely is hard and has attracted a fair amount of scholarly debate, the first major modern attempt to define the riddle was by Robert Petsch in 1899, with another seminal contribution, inspired by structuralism, by Robert A. Georges and Alan Dundes in 1963. In some traditions and contexts, riddles may overlap with proverbs, an example from a different language, Nothing hurts it, but it groans all the time can be deployed as a proverb or as a riddle. Much academic research on riddles has focused on collecting, cataloguing, defining and typologising riddles, key work on cataloguing and typologising riddles was published by Antti Aarne in 1918-20, and by Archer Taylor. In the case of ancient riddles recorded without solutions, considerable scholarly energy also goes into proposing and debating solutions, however, wide-ranging studies of riddles have tended to be limited to Western countries, with Oriental and African riddles being relatively neglected. Riddles have also attracted linguists, often studying riddles from the point of view of semiotics, the riddle was at times a prominent literary form in the ancient and medieval world, and so riddles are extensively, if patchily, attested in our written records from these periods. According to Archer Taylor, the oldest recorded riddles are Babylonian school texts which show no literary polish and it is clear that we have here riddles from oral tradition that a teacher has put into a schoolbook. It is thought that the worlds earliest surviving poetic riddles survive in the Sanskrit Rigveda, the first book of the Rigveda contains a number of riddles, overlapping in significant part with a collection of forty-seven in the Atharvaveda, riddles also appear elsewhere in Vedic texts. The highly sophisticated quality of many Sanskrit riddles can perhaps be illustrated by one rather simple example. Who makes a noise on seeing a thief, who is the enemy of lotuses. Who is the climax of fury, the answers to the first three questions, when combined in the manner of a charade, yield the answer to the fourth question. The first answer is bird, the dog, the third sun, and the whole is Viçvamitra, Ramas first teacher and counselor. Thus, for example, Daṇḍin cites this as an example of a name-riddle, A city, five letters, the one is a nasal. The Mahabharata also portrays riddle-contests and includes riddles accordingly, the first riddle collection in a medieval Indic language was by Amir Khusro, although he mostly wrote in Persian, he wrote his riddles in the language he called Hindawi. The riddles are in Mātrika metre, one example is, The emboldened text here indicates a clue woven into the text, it is a pun on nadi. While riddles are not numerous in the Bible, they are present, most famously in Samsons riddle in Judges xiv.14, but also in I Kings 10, 1-13, sirach also mentions riddles as a popular dinner pastime. The Aramaic Story of Ahikar contains a section of proverbial wisdom that in some versions also contains riddles. However, under the influence of Arabic literature in medieval al-Andalus, dunash ben Labrat, credited with transposing Arabic metres into Hebrew, composed a number of riddles, mostly apparently inspired by folk-riddles
4.
Logic puzzle
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A logic puzzle is a puzzle deriving from the mathematics field of deduction. The logic puzzle was first produced by Charles Lutwidge Dodgson, who is known under his pen name Lewis Carroll. Puzzles like this, where we are given a list of premises, Dodgson goes on to construct much more complex puzzles consisting of up to 8 premises. In the second half of the 20th century mathematician Raymond M. Smullyan has continued and expanded the branch of logic puzzles with books such as The Lady or the Tiger, to Mock a Mockingbird and Alice in Puzzle-Land. He popularized the knights and knaves puzzles, which involve knights, who tell the truth, and knaves. There are also puzzles that are completely non-verbal in nature. These are often referred to as logic grid puzzles, the most famous example may be the so-called Zebra Puzzle, which asks the question Who Owned the Zebra. For example, a map of a town might be present in lieu of a grid in a puzzle about the location of different shops. The dutch name for such a puzzle is logigram Category, Logic puzzles
5.
Dissection puzzle
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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, 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. Dissection puzzles are a form of geometric puzzle. The earliest known descriptions of puzzles are from the time of Plato in Ancient Greece. Other ancient dissection puzzles were used as graphic depictions of the Pythagorean theorem, in the 10th century, Arabic mathematicians used geometric dissections in their commentaries on Euclids Elements. In the 18th century, Chinese scholar Tai Chen described an elegant dissection for approximating the value of π, the puzzles saw a major increase in general popularity in the late 19th century when newspapers and magazines began running dissection puzzles. Puzzle creators Sam Loyd in the United States and Henry Dudeney in the United Kingdom were among the most published, the dissections of regular polygons and other simple geometric shapes into another such shape was the subject of Martin Gardners November 1961 Mathematical Games column in Scientific American. The haberdashers problem shown in the figure shows how to divide up a square. The column included a table of such best known dissections involving the square, pentagon, hexagon, greek cross, some types of dissection puzzle are intended to create a large number of different geometric shapes. The tangram is a dissection puzzle of this type. Some geometric forms are easy to create, while others present an extreme challenge and this variability has ensured the puzzles popularity. Other dissections are intended to move between a pair of geometric shapes, such as a triangle to a square, or a square to a five-pointed star, a dissection puzzle of this description is the haberdashers problem, proposed in 1907 by Henry Dudeney. The puzzle is a dissection of a triangle to a square and it is one of the simplest regular polygon to square dissections known, and is now a classic example. It is not known whether a dissection of a triangle to a square is possible with three pieces. Ostomachion Pizza theorem Puzzle Coffin, Stewart T, the Puzzling World of Polyhedral Dissections. Frederickson, Greg N. Dissections, Plane and Fancy, frederickson, Greg N. Piano-hinged Dissections, Time to Fold
6.
Mechanical puzzle
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A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces. The oldest known mechanical puzzle comes from Greece and appeared in the 3rd century BC, the game consists of a square divided into 14 parts, and the aim was to create different shapes from these pieces. In Iran “puzzle-locks” were made as early as the 17th century AD, the next known occurrence of puzzles is in Japan. In 1742 there is a mention of a game called “Sei Shona-gon Chie No-Ita” in a book, around the year 1800 the Tangram puzzle from China became popular, and 20 years later it had spread through Europe and America. The company Richter from Rudolstadt began producing large amounts of Tangram-like puzzles of different shapes, in 1893, Angelo John Lewis, using the pen name Professor Hoffman, wrote a book called Puzzles, Old and New. It contained, amongst other things, more than 40 descriptions of puzzles with secret opening mechanisms and this book grew into a reference work for puzzle games and modern copies exist for those interested. The beginning of the 20th century was a time in which puzzles were greatly fashionable, the puzzle shown in the picture, made of 12 identical pieces by W. Altekruse in the year 1890, was an example of this. With the invention of modern manufacture of many puzzles became easier and cheaper. In this category, the puzzle is present in component form, the Soma cube made by Piet Hein, the Pentomino by Solomon Golomb and the aforementioned laying puzzles Tangram and “Anker-puzzles” are all examples of this type of puzzle. Furthermore, problems in which a number of pieces have to be arranged so as to fit into a box are also classed in this category, the image shows a variant of Hoffmans packing problem. Modern tools such as laser cutters allow the creation of complex two-dimensional puzzles made of wood or acrylic plastic, in recent times this has become predominant and puzzles of extraordinarily decorative geometry have been designed. This makes use of the multitude of ways of subdividing areas into repeating shapes, computers aid in the design of new puzzles. A computer allows a search for solution – with its help a puzzle may be designed in such a way that it has the fewest possible solutions. The consequence is that solving the puzzle can be very difficult, the use of transparent materials enables the creation of puzzles, in which pieces have to be stacked on top of each other. The aim is to create a pattern, image or colour scheme in the solution. For example, one consists of several discs in which angular sections of varying sizes are differently coloured. The discs have to be stacked so as to create a circle around the discs. The puzzles in this category are usually solved by opening or dividing them into pieces and this includes those puzzles with secret opening mechanisms, which are to be opened by trial and error
7.
Disentanglement puzzle
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A disentanglement puzzle is a type of mechanical puzzle that involves disentangling one piece or set of pieces from another piece or set of pieces. The reverse problem of reassembling the puzzle can be as hard as—or even harder than—disentanglement, there are several different kinds of disentanglement puzzles, though a single puzzle may incorporate several of these features. A plate-and-ring puzzle usually consists of three pieces, one plate or similar displaying many holes and/or indentations a closed or nearly closed ring or a similar item, the plate as well as the ring are usually made from metal. The ring has to be disentangled from the plate, wire puzzles consist of two or more entangled pieces of more or less stiff wire. The pieces may or may not be closed loops, the closed pieces might be simple rings or have more complex shapes. Normally the puzzle must be solved by disentangling the two pieces without bending or cutting the wires, early wire puzzles were made from horseshoes and similar material. Wire-and-string puzzles usually consist of, one piece of string, ribbon or similar, one or several pieces of stiff wire sometimes additional pieces like wooden ball through which the string is threaded. One can distinguish three subgroups of wire-and-string puzzles, Closed string subgroup, Here the pieces of string consist of one closed loop, usually the string has to be disentangled from the wire. Unclosed loose string subgroup, Here the pieces of string are not closed, in this case the ends of the string are fitted with a ball, cube or similar which stops the string from slipping out too easily. Usually the string has to be disentangled from the wire, sometimes other tasks have to be completed instead, such as shifting a ring or ball from one end of the string to another end. Unclosed fixed string subgroup, Here the pieces of string are not closed, in these puzzles the string is not to be disentangled from the wire. One possible task may be to shift a ring or ball from one end of the string to another end
8.
Tiling puzzle
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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. Some tiling puzzles ask you to dissect a given shape first, 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, dissection puzzle Polyforms Sliding puzzle Tessellation Wang tile
9.
Sliding puzzle
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A sliding puzzle, sliding block puzzle, or sliding tile puzzle is a combination puzzle that challenges a player to slide pieces along certain routes to establish a certain end-configuration. The pieces to be moved may consist of shapes, or they may be imprinted with colors, patterns, sections of a larger picture, numbers. Sliding puzzles are essentially two-dimensional in nature, even if the sliding is facilitated by mechanically interlinked pieces or three-dimensional tokens, as this example shows, some sliding puzzles are mechanical puzzles. However, the fixtures are usually not essential to these puzzles. Unlike other tour puzzles, a block puzzle prohibits lifting any piece off the board. This property separates sliding puzzles from rearrangement puzzles, hence, finding moves and the paths opened up by each move within the two-dimensional confines of the board are important parts of solving sliding block puzzles. Chapmans invention initiated a puzzle craze in the early 1880s, from the 1950s through the 1980s sliding puzzles employing letters to form words were very popular. These sorts of puzzles have several possible solutions, as may be seen from such as Ro-Let, Scribe-o. The fifteen puzzle has been computerized and examples are available to play for free on-line from many Web pages and it is a descendant of the jigsaw puzzle in that its point is to form a picture on-screen. The last square of the puzzle is displayed automatically once the other pieces have been lined up. Winning Ways The 15 Puzzle US Patent 4872682 - sliding puzzle wrapped on Rubiks Cube
10.
Chess problem
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A chess problem, also called a chess composition, is a puzzle set by somebody using chess pieces on a chess board, that presents the solver with a particular task to be achieved. For instance, a position might be given with the instruction that White is to move first, a person who creates such problems is known as a composer. There is a deal of specialized jargon used in connection with chess problems. The term chess problem is not sharply defined, there is no clear demarcation between chess compositions on the one hand and puzzles or tactical exercises on the other, in practice, however, the distinction is very clear. There are common characteristics shared by compositions in the section of chess magazines, in specialist chess problem magazines. There is a stipulation, that is, a goal to be achieved, for example. There is a theme that the problem has been composed to illustrate, the problem exhibits economy in its construction, no greater force is employed than that required to render the problem sound. Problems are experienced not only as puzzles but as objects of beauty and this is closely related to the fact that problems are organised to exhibit clear ideas in as economical a manner as possible. Problems can be contrasted with tactical puzzles often found in columns or magazines in which the task is to find the best move or sequence of moves from a given position. Such puzzles are often taken from games, or at least have positions which look as if they could have arisen during a game. Most such puzzles fail to exhibit the above features, there are various different types of chess problems, Directmates, White to move first and checkmate Black within a specified number of moves against any defence. These are often referred to as mate in n, where n is the number of moves within which mate must be delivered. In composing and solving competitions, directmates are further broken down into three classes, Two-movers, White to move and checkmate Black in two moves against any defence, three-movers, White to move and checkmate Black in no more than three moves against any defence. More-movers, White to move and checkmate Black in n moves against any defence, helpmates, Black to move first cooperates with White to get Blacks own king mated in a specified number of moves. Selfmates, White moves first and forces Black to checkmate White, helpselfmates, White to move first cooperates with Black to get a position of selfmate in one move. Reflexmates, a form of selfmate with the stipulation that each side must give mate if it is able to do so. Seriesmovers, one makes a series of moves without reply to achieve a stipulated aim. Check may not be given except on the last move, a seriesmover may take various forms, Seriesmate, a directmate with White playing a series of moves without reply to checkmate Black
11.
Maze
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A maze is a path or collection of paths, typically from an entrance to a goal. The pathways and walls in a maze are typically fixed, but puzzles in which the walls, 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. Indoors, mirror mazes are another form of maze, in many of the apparent pathways are imaginary routes seen through multiple reflections in mirrors. Another type of maze consists of a set of rooms linked by doors, 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, mazes can be built with snow. Maze generation is the act of designing the layout of passages, there are many different approaches to generating mazes, with various maze generation algorithms for building them, either by hand or automatically by computer. There are two main mechanisms used to generate mazes, in carving passages, one marks out the network of available routes. In building a maze by adding walls, one lays out a set of obstructions within an open area, most mazes drawn on paper are done by drawing the walls, with the spaces in between the markings composing the passages. Maze solving is the act of finding a route through the maze from the start to finish, the mathematician Leonhard Euler was one of the first to analyze plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology. Mazes containing no loops are known as standard, or perfect mazes, thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, mazes are often used in psychology experiments to study spatial navigation and learning. Such experiments typically use rats or mice, examples are, Barnes maze Morris water maze Oasis maze Radial arm maze Elevated plus maze T-maze Ball-in-a-maze puzzles Dexterity puzzles which involve navigating a ball through a maze or labyrinth. Block maze A maze in which the player must complete or clear the maze pathway by positioning blocks, blocks may slide into place or be added. Linear or railroad maze A maze in which the paths are laid out like a railroad with switches, solvers are constrained to moving only forward. Often, a railroad maze will have a track for entrance. Logic mazes These are like standard mazes except they use other than dont cross the lines to restrict motion. Loops and traps maze A maze that features one-way doors, the doors can lead to the correct path or create traps that divert you from the correct path and lead you to the starting point. The player may not return through a door through which he has entered, the path is a series of loops interrupted by doors
12.
Word game
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Word games are spoken or board games often designed to test ability with language or to explore its properties. Word games are generally engaged as a source of entertainment, but have found to serve an educational purpose as well. For instance, young children can find enjoyment playing modestly competitive games such as Hangman, solving crossword puzzles, which requires familiarity with a larger vocabulary, is a pastime that mature adults have long credited with keeping their minds sharp. There are popular televised word games with valuable prizes for the winning contestants. Many word games enjoy international popularity across a multitude of languages, a word game or word puzzle can be of several different types, The goal is to form words out of given letters. Charades Fictionary LINQ Mad Libs Scattergories Taboo These are games based on words and letters
13.
Crossword
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A crossword is a word puzzle that normally takes the form of a square or a rectangular grid of white and black shaded squares. The goal is to fill the white squares with letters, forming words or phrases, in languages that are written left-to-right, the answer words and phrases are placed in the grid from left to right and from top to bottom. The shaded squares are used to separate the words or phrases, Crossword grids such as those appearing in most North American newspapers and magazines feature solid areas of white squares. Every letter is checked and usually each answer must contain at least three letters, in such puzzles shaded squares are typically limited to about one-sixth of the total. For example, if the top row has an answer running all the way across, another tradition in puzzle design is that the grid should have 180-degree rotational symmetry, so that its pattern appears the same if the paper is turned upside down. Most puzzle designs also require that all white cells be orthogonally contiguous, the design of Japanese crossword grids often follows two additional rules, that shaded cells may not share a side and that the corner squares must be white. The Swedish-style grid uses no numbers, as the clues are contained in the cells which do not contain answers. Arrows indicate in which direction the clues have to be answered and this style of grid is also used in several countries other than Sweden, often in magazines, but also in daily newspapers. These puzzles usually have no symmetry in the grid but instead often have a common theme Substantial variants from the forms exist. Two of the ones are barred crosswords, which use bold lines between squares to separate answers, and circular designs, with answers entered either radially or in concentric circles. Free form crosswords, which have simple, asymmetric designs, are seen on school worksheets, childrens menus. Grids forming shapes other than squares are also occasionally used, Puzzles are often one of several standard sizes. For example, many weekday newspaper puzzles are 15×15 squares, while weekend puzzles may be 21×21, 23×23 and their larger Sunday puzzle is about the same level of difficulty as a weekday-size Thursday puzzle. One of the smallest crosswords in general distribution is a 4×4 crossword compiled daily by John Wilmes, distributed online by USA Today as QuickCross and by Universal Uclick as PlayFour. Typically clues appear outside the grid, divided into an Across list and a Down list, for example, the answer to a clue labeled 17 Down is entered with the first letter in the cell numbered 17, proceeding down from there. Numbers are almost never repeated, numbered cells are numbered consecutively, usually left to right across each row, starting with the top row. Some Japanese crosswords are numbered top to bottom down each column, starting with the leftmost column. This ensures a proper name can have its initial capital letter checked with a letter in the intersecting clue
14.
Sudoku
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Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, the puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution. Completed games are always a type of Latin square with a constraint on the contents of individual regions. For example, the same single integer may not appear twice in the row, column. French newspapers featured variations of the puzzles in the 19th century, however, the modern Sudoku only started to become mainstream in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning single number. It first appeared in a US newspaper and then The Times in 2004, from the efforts of Wayne Gould, Number puzzles appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from magic squares. Le Siècle, a Paris daily, published a partially completed 9×9 magic square with 3×3 subsquares on November 19,1892, on July 6,1895, Le Siècles rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row, column, and broken diagonals contained only the numbers 1–9, although they are unmarked, each 3×3 subsquare does indeed comprise the numbers 1–9 and the additional constraint on the broken diagonals leads to only one solution. These weekly puzzles were a feature of French newspapers such as LEcho de Paris for about a decade, but disappeared about the time of World War I. Garnss name was present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon, whether or not Garns was familiar with any of the French newspapers listed above is unclear. At a later date, the name was abbreviated to Sudoku by Maki Kaji, Sudoku is a registered trademark in Japan and the puzzle is generally referred to as Number Place or, more informally, a portmanteau of the two words, Num Pla. In 1986, Nikoli introduced two innovations, the number of givens was restricted to no more than 32, and puzzles became symmetrical and it is now published in mainstream Japanese periodicals, such as the Asahi Shimbun. Gould devised a program to produce unique puzzles rapidly. Although the 9×9 grid with 3×3 regions is by far the most common, the Times offers a 12×12-grid Dodeka Sudoku with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 Number Place Challenger puzzles, Nikoli offers 25×25 Sudoku the Giant behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010, another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often, the limit takes the form of an extra dimension, the aforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in The Daily Mail, which use 6×6 grids
15.
Puzzle video game
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Puzzle video games are a genre of video games that emphasize puzzle solving. The types of puzzles can test many problem-solving skills including logic, pattern recognition, sequence solving, the player may have unlimited time or attempts to solve a puzzle, or there may be simple puzzles made difficult by having to complete them in real-time, as in Tetris. Puzzle games focus on logical and conceptual challenges, although often the games add time-pressure or other action-elements, although many action games and adventure games involve puzzles such as obtaining inaccessible objects, a true puzzle game focuses on puzzle solving as the primary gameplay activity. Games usually involve shapes, colors, or symbols, and the player must directly or indirectly manipulate them into a specific pattern. Rather than presenting a collection of puzzles to solve, puzzle games typically offer a series of related puzzles that are a variation on a single theme. This theme could involve pattern recognition, logic, or understanding a process and these games usually have a simple set of rules, where players manipulate game pieces on a grid, network or other interaction space. Players must unravel clues in order to achieve some victory condition, completing each puzzle will usually lead to a more difficult challenge, although some games avoid exhausting the player by offering easier levels between more difficult ones. There is a variety of puzzle games. Some feed to the player an assortment of blocks or pieces that they must organize in the correct manner, such as Tetris, Klax. Others present a game board or pieces and challenge the player to solve the puzzle by achieving a goal. Puzzle games are easy to develop and adapt, being implemented on dedicated arcade units, home video game consoles, personal digital assistants. An action puzzle or arcade puzzle requires that the player manipulates game pieces in an environment, often on a single screen and with a time limit. This is a term that has been used to describe several subsets of puzzle game. Firstly, it includes falling-block puzzles such as Tetris and KLAX and it includes games with characters moving through an environment, controlled either directly or indirectly. This can cross-over with other genres, a platform game which requires a novel mechanic to complete levels might be a puzzle platformer. Finally, it includes other action games that require timing and accuracy with pattern-matching or logic skills, other notable action puzzle games include Team Icos Ico and Shadow of the Colossus. A hidden object game is a genre of video game in which the player must find items from a list that are hidden within a picture. Hidden object games are a popular trend in gaming, and are comparatively inexpensive to buy
16.
Optical illusion
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An optical illusion is an illusion caused by the visual system and characterized by visually perceived images that differ from objective reality. The information gathered by the eye is processed in the brain to give a percept that does not tally with a measurement of the stimulus source. Pathological visual illusions arise from a pathological exaggeration in physiological visual perception mechanisms causing the aforementioned types of illusions, optical illusions are often classified into categories including the physical and the cognitive or perceptual, and contrasted with optical hallucinations. The Hermann grid illusion and Mach bands are two illusions that are best explained using a biological approach, once a receptor is active, it inhibits adjacent receptors. This inhibition creates contrast, highlighting edges, in the Hermann grid illusion the gray spots appear at the intersection because of the inhibitory response which occurs as a result of the increased dark surround. Lateral inhibition has also used to explain the Hermann grid illusion. More recent empirical approaches to optical illusions have had success in explaining optical phenomena with which theories based on lateral inhibition have struggled. A pathological visual illusion is a distortion of an external stimulus and are often diffuse. Pathological visual illusions usually occur throughout the field, suggesting global excitability or sensitivity alterations. Alternatively visual hallucination is the perception of a visual stimulus where none exists. Visual hallucinations are often from focal dysfunction and are usually transient and these symptoms may indicate an underlying disease state and necessitate seeing a medical practitioner. Etiologies associated with visual illusions include multiple types of ocular disease, migraines, hallucinogen persisting perception disorder, head trauma. If the visual illusions are diffuse and persistent, they affect the patients quality of life. These symptoms are refractory to treatment and may be caused by any of the aforementioned etiologes. There is no treatment for these visual disturbances. Cognitive illusions are commonly divided into ambiguous illusions, distorting illusions, paradox illusions, ambiguous illusions are pictures or objects that elicit a perceptual switch between the alternative interpretations. The Necker cube is an example, another instance is the Rubin vase. Distorting or geometrical-optical illusions are characterized by distortions of size, length, a striking example is the Café wall illusion
17.
Packing problems
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Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real life packaging, storage, each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a Bin packing problem, you are given, containers A set of some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or an object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps between goods and other goods or the container walls, in some variants, the aim is to find the configuration that packs a single container with the maximal density. More commonly, the aim is to all the objects into as few containers as possible. In some variants the overlapping is allowed but should be minimized, many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra and these problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, the counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one. That is, there always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency, in three dimensions, the face-centered cubic lattice offers the best lattice packing of spheres, and is believed to be the optimal of all packings. With simple sphere packings in three dimensions there are nine possible defineable packings, the 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. Cubes can easily be arranged to fill space completely, the most natural packing being the cubic honeycomb. No other Platonic solid can tile space on its own, tetrahedra can achieve a packing of at least 85%. One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic lattice, tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb
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Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic
19.
Syllogism
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A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a statement and a specific statement. For example, knowing that all men are mortal and that Socrates is a man, Syllogistic arguments are usually represented in a three-line form, All men are mortal. In antiquity, two theories of the syllogism existed, Aristotelian syllogistic and Stoic syllogistic. Aristotle defines the syllogism as. a discourse in which certain things having been supposed, despite this very general definition, in Aristotles work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably and this article is concerned only with this traditional use. The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle, the onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle develops his theory of the syllogism. Prior Analytics, upon re-discovery, was regarded by logicians as a closed and complete body of doctrine, leaving very little for thinkers of the day to debate. Aristotles theories on the syllogism for assertoric sentences was considered especially remarkable, Aristotles Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise. Aristotles terminology in this aspect of his theory was deemed vague and in many cases unclear and his original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, boethius contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the twelfth century and his perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelards distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a coherent concept of Aristotles modal syllogism model. For two hundred years after Buridans discussions, little was said about syllogistic logic, the Aristotelian syllogism dominated Western philosophical thought for many centuries. In the 17th century, Sir Francis Bacon rejected the idea of syllogism as being the best way to draw conclusions in nature. Instead, Bacon proposed a more inductive approach to the observation of nature, in the 19th century, modifications to syllogism were incorporated to deal with disjunctive and conditional statements. Kant famously claimed, in Logic, that logic was the one completed science, though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kants opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift. This introduced a calculus, a method of representing categorical statements by the use of quantifiers, in the last 20 years, Bolzanos work has resurfaced and become subject of both translation and contemporary study
20.
Thinking outside the box
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Thinking outside the box entails a thinking process, which comprehends the implementation of an unusual approach to the logical thinking structure. It´s a procedure which aims to escape relational reasoning and thinking, thinking outside the box is a metaphor that means to think differently, unconventionally, or from a new perspective. This phrase often refers to novel or creative thinking, the term is thought to derive from management consultants in the 1970s and 1980s challenging their clients to solve the nine dots puzzle, whose solution requires some lateral thinking. The catchphrase, or cliché, has widely used in business environments, especially by management consultants and executive coaches. To think outside the box is to further and to try not thinking of the obvious things. A simplified definition for paradigm is a habit of reasoning or a conceptual framework, a simplified analogy is the box in the commonly used phrase thinking outside the box. What is encompassed by the words inside the box is analogous with the current, creative thinking acknowledges and rejects the accepted paradigm to come up with new ideas. The notion of something outside a box is related to a traditional topographical puzzle called the nine dots puzzle. The origins of the phrase thinking outside the box are obscure, but it was popularized in part because of a nine-dot puzzle, the nine dots puzzle is much older than the slogan. It appears in Sam Loyds 1914 Cyclopedia of Puzzles, in the 1951 compilation The Puzzle-Mine, Puzzles Collected from the Works of the Late Henry Ernest Dudeney, the puzzle is attributed to Dudeney himself. Sam Loyds original formulation of the puzzle entitled it as Christopher Columbuss egg puzzle and this was an allusion to the story of Egg of Columbus. The puzzle proposed an intellectual challenge—to connect the dots by drawing four straight, continuous lines that pass each of the nine dots. The conundrum is easily resolved, but only by drawing the lines outside the confines of the area defined by the nine dots themselves. The phrase thinking outside the box is a restatement of the solution strategy, the puzzle only seems difficult because people commonly imagine a boundary around the edge of the dot array. The heart of the matter is the barrier that people typically perceive. Ironically, telling people to think outside the box does not help them think outside the box and this is due to the distinction between procedural knowledge and declarative knowledge. For example, a cue such as drawing a square outside the 9 dots does allow people to solve the 9-dot problem better than average. The nine-dot problem is a well-defined problem and it has a clearly stated goal, and all necessary information to solve the problem is included
21.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
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Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
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Operation (mathematics)
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In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation, the most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant, the mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered, in this context, the usual operations, of finite arity are also called finitary operations. There are two types of operations, unary and binary. Unary operations involve only one value, such as negation and trigonometric functions, binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation. Operations can involve mathematical objects other than numbers, the logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted, rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the operation of complementation. Operations on functions include composition and convolution, operations may not be defined for every possible value. For example, in the numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain, the set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the numbers, the squaring operation only produces non-negative numbers. A vector can be multiplied by a scalar to form another vector, and the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, the values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs, an operation is like an operator, but the point of view is different. An operation ω is a function of the form ω, V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, thus a unary operation has arity one, and a binary operation has arity two
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Combination
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In mathematics, a combination is a way of selecting items from a collection, such that the order of selection does not matter. In smaller cases it is possible to count the number of combinations, more formally, a k-combination of a set S is a subset of k distinct elements of S. The set of all k-combinations of a set S is sometimes denoted by, combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, k-multiset, or k-combination with repetition are often used. If, in the example, it was possible to have two of any one kind of fruit there would be 3 more 2-selections, one with two apples, one with two oranges, and one with two pears. Although the set of three fruits was small enough to write a complete list of combinations, with large sets this becomes impractical, for example, a poker hand can be described as a 5-combination of cards from a 52 card deck. The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter, there are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 /2,598,960. The same number however occurs in other mathematical contexts, where it is denoted by, notably it occurs as a coefficient in the binomial formula. One can define for all natural numbers k at once by the relation n = ∑ k ≥0 X k, binomial coefficients can be computed explicitly in various ways. To get all of them for the expansions up to n, one can use the recursion relation = +, for 0 < k < n, which follows from n = n −1, this leads to the construction of Pascals triangle. For determining an individual binomial coefficient, it is practical to use the formula = n ⋯ k. When k exceeds n/2, the formula contains factors common to the numerator and the denominator. This expresses a symmetry that is evident from the formula, and can also be understood in terms of k-combinations by taking the complement of such a combination. Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember and it is obtained from the previous formula by multiplying denominator and numerator by. So it is inferior as a method of computation to that formula. The last formula can be directly, by considering the n. permutations of all the elements of S. Each such permutation gives a k-combination by selecting its first k elements, =52 ×51 ×50 ×49 ×48 ×47. Another alternative computation, equivalent to the first, is based on writing =1 ×2 ×3 × ⋯ × k, which gives =521 ×512 ×503 ×494 ×485 =2,598,960
25.
Rubik's Cube
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Rubiks Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. As of January 2009,350 million cubes had been sold making it the worlds top-selling puzzle game. It is widely considered to be the worlds best-selling toy, in a classic Rubiks Cube, each of the six faces is covered by nine stickers, each of one of six solid colours, white, red, blue, orange, green, and yellow. In currently sold models, white is opposite yellow, blue is green, and orange is opposite red. On early cubes, the position of the colours varied from cube to cube, an internal pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be returned to have one colour. Similar puzzles have now produced with various numbers of sides, dimensions. Although the Rubiks Cube reached its height of popularity in the 1980s, it is still widely known. Many speedcubers continue to practice it and other twisty puzzles and compete for the fastest times in various categories, since 2003, The World Cube Association, the Rubiks Cubes international governing body, has organised competitions worldwide and kept the official world records. In March 1970, Larry D. Nichols invented a 2×2×2 Puzzle with Pieces Rotatable in Groups, Nicholss cube was held together with magnets. Patent 3,655,201 on April 11,1972, on April 9,1970, Frank Fox applied to patent his Spherical 3×3×3. He received his UK patent on January 16,1974, in the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. He did not realise that he had created a puzzle until the first time he scrambled his new Cube, Rubik obtained Hungarian patent HU170062 for his Magic Cube in 1975. Rubiks Cube was first called the Magic Cube in Hungary, Ideal wanted at least a recognisable name to trademark, of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980. The first test batches of the Magic Cube were produced in late 1977, Magic Cube was held together with interlocking plastic pieces that prevented the puzzle being easily pulled apart, unlike the magnets in Nicholss design. With Ernő Rubiks permission, businessman Tibor Laczi took a Cube to Germanys Nuremberg Toy Fair in February 1979 in an attempt to popularise it. It was noticed by Seven Towns founder Tom Kremer and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide, the puzzle made its international debut at the toy fairs of London, Paris, Nuremberg and New York in January and February 1980. After its international debut, the progress of the Cube towards the toy shop shelves of the West was briefly halted so that it could be manufactured to Western safety, a lighter Cube was produced, and Ideal decided to rename it
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Virtual reality
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VR also simulates a users physical presence in this environment. A person using virtual reality equipment is able to look around the world, and with high quality VR move about in it. Virtual reality is displayed with a virtual reality headset, VR headsets are head-mounted goggles with a screen in front of the eyes. Programs may include audio and sounds through speakers or headphones, advanced haptic systems may include tactile information, generally known as force feedback in medical, video gaming and military training applications. Some VR systems used in games can transmit vibrations and other sensations to the user through the game controller. Virtual reality also refers to remote communication environments which provide a presence of users with through telepresence and telexistence or the use of a virtual artifact. The immersive environment can be similar to the world in order to create a life-like experience grounded in reality or sci-fi. In 1938, Antonin Artaud described the nature of characters and objects in the theatre as la réalité virtuelle in a collection of essays. The English translation of book, published in 1958 as The Theater. The term artificial reality, coined by Myron Krueger, has been in use since the 1970s, the term virtual reality was used in The Judas Mandala, a 1982 science fiction novel by Damien Broderick. Virtual has had the meaning being something in essence or effect, the term virtual has been used in the computer sense of not physically existing but made to appear by software since 1959. A dictionary definition for cyberspace states that word is a synonym for virtual reality. Virtual reality shares some elements with augmented reality, AR is a type of virtual reality technology that blends what the user sees in their real surroundings with digital content generated by computer software. The additional software-generated images with the virtual scene typically enhance way the real look in some way. Some AR systems use a camera to capture the surroundings or some type of display screen which the user looks at. The Virtual Reality Modelling Language, first introduced in 1994, was intended for the development of worlds without dependency on headsets. The Web3D consortium was founded in 1997 for the development of industry standards for web-based 3D graphics. The consortium subsequently developed X3D from the VRML framework as an archival and these components led to relative affordability for independent VR developers, and lead to the 2012 Oculus Rift kickstarter offering the first independently developed VR headset
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Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
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N-dimensional sequential move puzzle
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The Rubiks Cube is the original and best known of the three-dimensional sequential move puzzles. There have been many implementations of this puzzle in software. It is an extension to create sequential move puzzles in more than three dimensions. Hence, they can be simulated by software, as with the mechanical sequential move puzzles, there are records for solvers, although not yet the same degree of competitive organisation. A zero-dimensional point at which higher-dimension figures meet, a one-dimensional figure at which higher-dimension figures meet. A two-dimensional figure at which higher-dimension figures meet, a three-dimensional figure at which higher-dimension figures meet. A n-dimensional figure continuing as above, a specific geometric shape may replace polytope where this is appropriate, such as 4-cube to mean the tesseract. A higher-dimension figure containing n cells, a single moveable part of the puzzle having the same dimensionality as the whole puzzle. In the solving community this is the generally used for a piece. The coloured labels on the puzzle which identify the state of the puzzle, for instance, the corner cubies of a Rubik cube are a single piece but each has three stickers. The stickers in higher-dimensional puzzles will have a dimensionality greater than two, for instance, in the 4-cube, the stickers are three-dimensional solids. For comparison purposes, the relating to the standard 33 Rubik cube is as follows. 2 ⋅2122 ⋅383 ∼1020 There is some debate whether the face-centre cubies should be counted as separate pieces as they cannot be moved relative to each other. A different number of pieces may be given in different sources, in this article the face-centre cubies are counted as this makes the arithmetical sequences more consistent and they can certainly be rotated, a solution of which requires algorithms. However, the right in the middle is not counted because it has no visible stickers. Arithmetically we should have P = V + E + F + C But P is always one short of this in the figures given in this article because C is not being counted. Geometric shape, tesseract The Superliminal MagicCube4D software implements many twisty puzzle versions of 4D polytopes including N4 cubes, the UI allows for 4D twists and rotations plus control of 4D viewing parameters such as the projection into 3D, cubie size and spacing, and sticker size. Superliminal Software maintains a Hall of Fame for record breaking solvers of this puzzle,2 ⋅223 ⋅31 ⋅3 ⋅15 ⋅4 ∼10120 Achievable combinations, =15
29.
Regular polyhedra
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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive, in classical contexts, many different equivalent definitions are used, a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form, there are 5 finite convex regular polyhedra, known as the Platonic solids. These are the, tetrahedron, cube, octahedron, dodecahedron and icosahedron, there are also four regular star polyhedra, making nine regular polyhedra in all. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons, All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre, An insphere, tangent to all faces, an intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices, the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them, Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry, the five Platonic solids have an Euler characteristic of 2. Some of the stars have a different value. The sum of the distances from any point in the interior of a polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra, in a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows, The tetrahedron is self-dual, the cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other, the small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other, the Schläfli symbol of the dual is just the original written backwards, for example the dual of is. See also Regular polytope, History of discovery, stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery, the earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, euclids reference to Plato led to their common description as the Platonic solids
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Semiregular polyhedron
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The term semiregular polyhedron is used variously by different authors. In its original definition, it is a polyhedron with faces and a symmetry group which is transitive on its vertices. These polyhedra include, The thirteen Archimedean solids, an infinite series of convex prisms. An infinite series of convex antiprisms and these semiregular solids can be fully specified by a vertex configuration, a listing of the faces by number of sides in order as they occur around a vertex. For example,3.5.3.5, represents the icosidodecahedron which alternates two triangles and two pentagons around each vertex,3.3.3.5 in contrast is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive, since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte provided a definition which Coxeter found too artificial, Coxeter himself dubbed Gossets figures uniform, with only a quite restricted subset classified as semiregular. Yet others have taken the path, categorising more polyhedra as semiregular. These include, Three sets of polyhedra which meet Gossets definition. The duals of the above semiregular solids, arguing that since the polyhedra share the same symmetries as the originals. These duals include the Catalan solids, the convex dipyramids and antidipyramids or trapezohedra, a further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing. Gossets definition of semiregular includes figures of higher symmetry, the regular and quasiregular polyhedra and this naming system works well, and reconciles many of the confusions. Assuming that ones stated definition applies only to convex polyhedra is probably the most common failing, Coxeter, Cromwell and Cundy & Rollett are all guilty of such slips. In many works semiregular polyhedron is used as a synonym for Archimedean solid and we can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular and facially-transitive duals. Later, Coxeter would quote Gossets definition without comment, thus accepting it by implication, peter Cromwell writes in a footnote to Page 149 that, in current terminology, semiregular polyhedra refers to the Archimedean and Catalan solids. On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans, by implication this treats the Catalans as not semiregular, thus effectively contradicting the definition he provided in the earlier footnote. Semiregular polytope Regular polyhedron Weisstein, Eric W. Semiregular polyhedron
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Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations
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Cuboid
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In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
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Rectilinear polygon
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A rectilinear polygon is a polygon all of whose edge intersections are at right angles. Thus the interior angle at each vertex is either 90° or 270°, rectilinear polygons are a special case of isothetic polygons. In many cases another definition is preferable, a polygon is a polygon with sides parallel to the axes of Cartesian coordinates. The distinction becomes crucial when spoken about sets of polygons, the definition would imply that sides of all polygons in the set are aligned with the same coordinate axes. Within the framework of the definition it is natural to speak of horizontal edges. Rectilinear polygons are also known as orthogonal polygons, other terms in use are iso-oriented, axis-aligned, and axis-oriented polygons. The importance of the class of rectilinear polygons comes from the following and they are convenient for the representation of shapes in integrated circuit mask layouts due to their simplicity for design and manufacturing. Many manufactured objects result in orthogonal polygons, problems in computational geometry stated in terms of polygons often allow for more efficient algorithms when restricted to orthogonal polygons. An example is provided by the art gallery theorem for orthogonal polygons, a rectilinear polygon has edges of two types, horizontal and vertical. Lemma, The number of edges is equal to the number of vertical edges. Corollary, Orthogonal polygons have a number of edges. A rectilinear polygon has corners of two types, corners in which the angle is interior to the polygon are called convex. A knob is an edge whose two endpoints are convex corners, an antiknob is an edge whose two endpoints are concave corners. A rectilinear polygon that is simple is also called hole-free because it has no holes - only a single continuous boundary. It has several interesting properties, The number of convex corners is four more than the number of concave corners, to see why, imagine that you traverse the boundary of the polygon clockwise. At a convex corner, you turn 90° right, at any concave corner, finally you must make an entire 360° turn and come back to the original point, hence the number of right turns must be 4 more than the number of left turns. Corollary, every rectilinear polygon has at least 4 convex corners, the number of knobs is four more than the number of antiknobs. To see why, let X be the number of convex corners and Y the number of concave corners. Let XX the number of convex corners followed by a corner, XY the number of convex corners followed by a concave corner, YX
