Disentanglement puzzles are 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 harder than—disentanglement. There are several different kinds of disentanglement puzzles, though a single puzzle may incorporate several of these features. Wire-and-string puzzles consist of: one piece of string, ribbon or similar, which may form a closed loop or which may have other pieces like balls fixed to its end. 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: The pieces of string consist of one closed loop, as in the Baguenaudier puzzle; the string has to be disentangled from the wire. Unclosed loose string subgroup: The pieces of string are not closed, are not attached to the wire. 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.
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: The pieces of string are not closed, but are somewhere on its length attached to the wire. In these puzzles the string is not to be disentangled from the wire. One possible task may be to shift a ball from one end of the string to another end. One difficult puzzle was designed by R. Boomhower in 1966 and has been modified into different designs. Different versions include a paddle-shaped design, a vertical beam on a wood support, two vertical beams on a wood support. Variations have the string passing through the slot once or two times. Names have included the Boomhower puzzle, T-Bar puzzle, Wit's End puzzle, the Mini Rope Bridge puzzle; some sources identify a topologically-equivalent puzzle called the Mystery Key issued by the Peter Pan company in the 1950s. Wire puzzles consist of more entangled pieces of more or less stiff wire.
The pieces may not be closed loops. The closed pieces might have more complex shapes; the puzzle must be solved by disentangling the two pieces without bending or cutting the wires. Early wire puzzles were made from similar material. A plate-and-ring puzzle 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 made from metal. The ring has to be disentangled from the plate; some puzzles have been created which may appear deceptively simple, but are impossible to solve. One such puzzle is the "Notorious Figure Eight Puzzle", it is sometimes sold with instructions giving hints as to its level of difficulty, a "solution" is provided but is vague and impossible to follow, but the puzzle is impossible to solve. Most puzzle solvers try to solve such puzzles by mechanical manipulation, but some branches of mathematics can be used to create a model of disentanglement puzzles. Applying a configuration space with a topological framework is an analytical method to gain insight into the properties and solution of some disentanglement puzzles.
However, some mathematicians have stated that capturing the important aspects of many such puzzles can be difficult, there is no universal algorithm that will provide the solution to such puzzles. Borromean rings, a method of linking three closed loops, found in some disentanglement puzzles Human knot Unknotting problem Unlink
A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces in which the solution is to manipulate the whole object or parts of it. One of the most well-known mechanical puzzles is Ernő Rubik’s Cube that he invented in 1974; the puzzles are designed for a single player where the goal is for the player to see through the principle of the object, not so much that they accidentally come up with the right solution through trial and error. With this in mind, they are used as an intelligence test or in problem solving training; 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, the aim was to create different shapes from these pieces; this is not easy to do. 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, 20 years it had spread through Europe and America.
The company Richter from Rudolstadt began producing large amounts of Tangram-like puzzles of different shapes, the so-called "Anker-puzzles" in about 1891. In 1893, Angelo John Lewis, using the pen name "Professor Hoffman", wrote, it contained, among other things, more than 40 descriptions of puzzles with secret opening mechanisms. 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 fashionable and the first patents for puzzles were recorded. With the invention of modern polymers manufacture of many puzzles became cheaper. In this category, the puzzle is present in component form, the aim is to produce a certain shape; 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 classed in this category.
The image shows a variant of Hoffman's packing problem. The aim is to pack 27 cuboids with side lengths A, B, C into a box of side length A+B+C, subject to two constraints: 1) A, B, C must not be equal 2) The smallest of A, B, C must be larger than / 4 One possibility would be A=18, B=20, C=22 – the box would have to have the dimensions 60×60×60. 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 an exhaustive search for solution – with its help a puzzle may be designed in such a way that it has the fewest possible solutions, or a solution requiring the most steps possible; the consequence is that solving the puzzle can be 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 specific image or colour scheme in the solution. For example, one puzzle 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 colour circle around the discs. The puzzles in this category are solved by opening or dividing them into pieces; this includes those puzzles with secret opening mechanisms, which are to be opened by trial and error. Furthermore, puzzles consisting of several metal pieces linked together in some fashion are considered part of this category; the two puzzles shown in the picture are good for social gatherings, since they appear to be easily taken apart, but in reality many people cannot solve this puzzle. The problem here lies in the shape of the interlocking pieces – the mating surfaces are tapered, thus can only be removed in one direction. However, each piece has two oppositely sloping tapers mating with the two adjoining pieces so that the piece cannot be removed in either direction.
Boxes called secret boxes or puzzle boxes with secret opening mechanisms popular in Japan, are included in this category. These caskets contain more or less complex invisible opening mechanisms which reveal a small hollow space on opening. There is a vast variety of opening mechanisms, such as hardly visible panels which need to be shifted, inclination mechanisms, magnetic locks, movable pins which need to be rotated into a certain position up and time locks in which an object has to be held in a given position until a liquid has filled up a certain container. In an interlocking puzzle, one or more pieces hold the rest together, or the pieces are mutually self-sustaining; the aim is to disassemble and reassemble the puzzle. Both assembly and disassembly can be difficult – contrary to assembly puzzles, these puzzles do not just fall apart easily; the level of difficulty is assessed in terms of the number of moves required to remove the first piece from the initial puzzle. Puzzles introduced elements of rotation.
The known history of these puzzles reaches back to the beginning of the 18th century. In 1803 a catalog by "Bastelmeier" contained two puzzles of this type. Professor Hoffman's puzzle book mentioned above contained two interlocking puzzles. At the beginning of the 19th century the Jap
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, or logic puzzles. Puzzles are created to be 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; 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, by Abram Kendall, master; the word came to be used as a noun. The word puzzle comes from pusle, meaning "bewilder, confound", a frequentive of the obsolete verb pose in the sense of "perplex".
The use of the word to mean "a toy contrived to test one's ingenuity" is recent. 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, disentanglement puzzles, lock puzzles, folding puzzles, combination puzzles, mechanical puzzles. A chess problem is a puzzle. Examples are the eight queens puzzle. Jigsaw puzzles. Lateral thinking puzzles called "situation puzzles" Mathematical puzzles include the missing square puzzle and many impossible puzzles — puzzles which have no solution, such as the Seven Bridges of Königsberg, the three cups problem, three utilities problem Mechanical puzzles such as the Rubik's Cube and Soma cube Metapuzzles are puzzles which unite elements of other puzzles. Paper-and-pencil puzzles such as Uncle Art's Funland, connect the dots, nonograms Also the logic puzzles published by Nikoli: Sudoku, Kakuro, Hashiwokakero, Hitori, Light Up, Number Link, Ripple Effect and Kuromasu.
Peg solitaire. Rubik's Cube and other combination puzzles can be stimulating toys for children or recreational activities for adults. Sangaku Sliding puzzles such as the 15 Puzzle. Puzz-3D is a three-dimensional variant of this type. Sokoban Spot the difference Tangram Word puzzles, including anagrams, crossword puzzles and word search puzzles. Tabletop and digital word puzzles include Bananagrams, Bonza, Letterpress, Puzzlage, Ruzzle, Upwords, WordSpot, Words with Friends. Wheel of Fortune is a game show centered on a word puzzle. Solutions of puzzles require the recognition of patterns and the adherence to a particular kind of ordering. 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 more by those with good deduction skills. Deductive reasoning improves with practice. Mathematical puzzles involves BODMAS. BODMAS is an acronym and it stands for Bracket, Of, Multiplication and Subtraction.
In certain regions, PEDMAS is the synonym of BODMAS. It explains the order of operations to solve an expression; some mathematical puzzle requires Top to Bottom convention to avoid the ambiguity in the order of operations. It is an elegantly simple idea that relies, as sudoku does, on the requirement that numbers appear only once starting from top to bottom as coming along. Puzzle makers are people; some notable creators of puzzles are: Ernő Rubik Sam Loyd Henry Dudeney Boris Kordemsky David J. Bodycombe Will Shortz Lloyd King Martin Gardner Raymond Smullyan Jigsaw puzzles are the most popular form of puzzle. Jigsaw puzzles were invented around 1760, when John Spilsbury, a British engraver and cartographer, mounted a map on a sheet of wood, which he sawed around the outline of each individual country on the map, he used the resulting pieces as an aid for 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.
The largest puzzle is made by German game company Ravensburger. The smallest puzzle made was created at LaserZentrum Hannover, it is the size of a sand grain. By the early 20th century and newspapers had found that they could increase their readership by publishing puzzle contests, beginning with crosswords and in modern days sudoku. There are organizations and events that cater to puzzle enthusiasts, such as: Nob Yoshigahara Puzzle Design Competition World Puzzle Championship National Puzzlers' League Puzzlehunts such as the Maze of Games List of impossible puzzles List of Nikoli puzzle types Riddle Puzzles at DMOZ
A рuzzlehunt is a puzzle game where teams compete to solve a series of puzzles at a particular site, in multiple sites or via the internet. Groups of puzzles in a puzzle hunt are connected by a metapuzzle, leading to answers which combine into a final set of solutions; some famous annual puzzlehunts are: D. A. S. H. Takes place on the same day in multiple cities around the world using ClueKeeper as the interface; the MIT Mystery Hunt, the Melbourne University Mathematics & Statistics Society puzzlehunt, the Sydney University Maths Society puzzlehunt, the TMOU the Microsoft Puzzle Hunt, the Miami Herald's Tropic Hunt, the Washington Post's Post Hunt, the Gen Con Puzzle Hunt, Galactic Puzzle Hunt, run over Pi Day weekend, The Phish.net Quest puzzle sequence, The Great Puzzle Hunt in Bellingham, WA every AprilCollege puzzlehunts include the aforementioned MIT Mystery Hunt as well as: PuzzleCrack College Puzzle Challenge, hosted by Microsoft as a recruiting event on college campuses. Palantir's Puzzle Challenge, a recruiting event on multiple campuses hosted by Palantir Technologies Google Games, a multi-part competition that includes logic puzzles, coding and building challenges that utilize materials like LEGO bricks APT Puzzle Tournament, a recruiting event on multiple campuses hosted by Applied Predictive Technologies Puzzle Hunt, put on every semester by a student organization called PuzzleHuntCMU at Carnegie Mellon University's Pittsburgh, PA campus Nova Quest, a campus-wide puzzlehunt organized by the Nova Quest student organization, taking place each spring at Villanova University Puzzle Hunt, open to all students and organized by the Rice University IEEE student chapter The VT Hunt, an annual puzzlehunt at Virginia Tech that involves both abstract puzzles and physical clues Puzzlehunt, a puzzlehunt made annually by the Stanford University Mathematical Organization Berkeley Mystery Hunt, a puzzlehunt at UC Berkeley made by The Campus League of PuzzlersSometimes, the prize for winning a puzzlehunt is to create the next one.
The Game Microsoft Puzzle Safari Race In The City Puzzled Pint Prehistoric Puzzlehunt www.priweb.org/puzzlehunt Mission Street Puzzles UNR Puzzle Hunt Puzzle Hunt Calendar Puzzle Pile Event Calendar Letterboxing Geocaching Alternate reality game Treasure hunt Geohashing Encounter La chouette d'or The Last of Sheila, a murder mystery film set at a puzzlehunt
A crossword is a word puzzle that takes the form of a square or a rectangular grid of white-and black-shaded squares. The game's goal is to fill the white squares with letters, forming words or phrases, by solving clues, which lead to the answers. 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 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 each answer must contain at least three letters. In such puzzles shaded squares are limited to about one-sixth of the total. Crossword grids elsewhere, such as in Britain, South Africa and Australia, have a lattice-like structure, with a higher percentage of shaded squares, leaving about half the letters in an answer unchecked. For example, if the top row has an answer running all the way across, there will be no across answers in the second row.
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 require that all white cells be orthogonally contiguous; the design of Japanese crossword grids 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 clue 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: horizontal; this style of grid is used in several countries other than Sweden in magazines, but in daily newspapers. The grid has one or more photos replacing a block of squares as a clue to one or several answers, for example, the name of a pop star, or some kind of rhyme or phrase that can be associated with the photo; these puzzles have no symmetry in the grid but instead have a common theme Substantial variants from the usual forms exist.
Two of the common ones are barred crosswords, which use bold lines between squares to separate answers, circular designs, with answers entered either radially or in concentric circles. "Free form" crosswords, which have simple, asymmetric designs, are seen on school worksheets, children's menus, other entertainment for children. Grids forming shapes other than squares are occasionally used. Puzzles are one of several standard sizes. For example, many weekday newspaper puzzles are 15×15 squares, while weekend puzzles may be 21×21, 23×23, or 25×25; the New York Times puzzles set a common pattern for American crosswords by increasing in difficulty throughout the week: their Monday puzzles are the easiest and the puzzles get harder each day until Saturday. Their larger Sunday puzzle is about the same level of difficulty as a weekday-size Thursday puzzle; this has led U. S. solvers to use the day of the week as a shorthand when describing how hard a puzzle is: e.g. an easy puzzle may be referred to as a "Monday" or a "Tuesday", a medium-difficulty puzzle as a "Wednesday", a difficult puzzle as a "Saturday".
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". 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 never repeated; some Japanese crosswords are numbered from top to bottom down each column, starting with the leftmost column and proceeding right. Capitalization of answer letters is conventionally ignored; this ensures a proper name can have its initial capital letter checked with a non-capitalizable letter in the intersecting clue. Diacritical markings in foreign loanwords are ignored for similar reasons; some crossword clues, called straight or quick clues, are simple definitions of the answers. Some clues may feature anagrams, these are explicitly described as such. A straight clue is not in itself sufficient to distinguish between several possible answers, either because multiple synonymous answers may fit or because the clue itself is a homonym, so the solver must make use of checks to establish the correct answer with certainty.
For example, the answer to the clue "PC key" for a three-letter answer could be ESC, ALT, TAB, DEL, or INS, so until a check is filled in, giving at least one of the letters, the correct answer cannot be determined. In most American-style crosswords, the majority of the clues in the puzzle are straight clues, with the remainder being one of the other types described be
A riddle is a statement or question or phrase having a double or veiled meaning, put forth as a puzzle to be solved. Riddles are of two types: enigmas, which are problems expressed in metaphorical or allegorical language that require ingenuity and careful thinking for their solution, conundra, which are questions relying for their effects on punning in either the question or the answer. Archer Taylor says that "we can say that riddling is a universal art" and cites riddles from hundreds of different cultures including Finnish, American Indian, Russian and Filipino sources amongst many others. Many riddles and riddle-themes are internationally widespread. However, at least in the West, if not more "riddles have in the past few decades ceased to be part of oral tradition", being replaced by other oral-literary forms, by other tests of wit such as quizzes. In the assessment of Elli Köngas Maranda, whereas myths serve to encode and establish social norms, "riddles make a point of playing with conceptual boundaries and crossing them for the intellectual pleasure of showing that things are not quite as stable as they seem" – though the point of doing so may still be to "play with boundaries, but to affirm them".
Defining riddles 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. Georges and Dundes suggested that "a riddle is a traditional verbal expression which contains one or more descriptive elements, a pair of which may be in opposition. There are many possible sub-sets of the riddle, including charades and some jokes. In some traditions and contexts, riddles may overlap with proverbs; the Russian phrase "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 and typologising riddles. Key work on cataloguing and typologising riddles was published by Antti Aarne in 1918–20, by Archer Taylor. In the case of ancient riddles recorded without solutions, considerable scholarly energy goes into proposing and debating solutions.
Whereas researchers had tended to take riddles out of their social performance contexts, the rise of anthropology in the post-War period encouraged more researchers to study the social role of riddles and riddling. However, wide-ranging studies of riddles have tended to be limited to Western countries, with Oriental and African riddles being neglected. Riddles have attracted linguists studying riddles from the point of view of semiotics. Many riddles appear in similar form across many countries, continents. Borrowing of riddles happens on a small, local scale, across great distances. Dorvlo gives an example of a riddle, borrowed from the Ewe language by speakers of the neighboring Logba language: "This woman has not been to the riverside for water, but there is water in her tank"; the answer is a coconut. On a much wider scale, the Riddle of the Sphinx has been documented in the Marshall Islands carried there by Western contacts in the last two centuries. Key examples of internationally widespread riddles, with a focus on European tradition, based on the classic study by Antti Aarne.
The basic form of this riddle is'White field, black seeds', where the field is a page and the seeds are letters. An example is the eighth- or ninth-century Veronese Riddle: Here, the oxen are the scribe's finger and thumb, the plough is the pen. Among literary riddles, riddles on the pen and other writing equipment are widespread; this type is found across Eurasia. For example, a riddle in the Sanskrit Rig Veda describes a'twelve-spoked wheel, upon which stand 720 sons of one birth'; the most famous example of this type is the Riddle of the Sphinx. This Estonian example shows the pattern: The riddle describes a crawling baby, a standing person, an old person with a walking stick; this type includes riddles along the lines of this German example: The conceit here is that Two-legs is a person, Three-legs is a three-legged stool, Four-legs is a dog, One-leg is a walking stick. An example of this type is given here in thirteenth-century Icelandic form: The cow has four udders, four legs, two horns, two back legs, one tail.
This is a French version of the type. An English version is: Here, a snowflake falls from the sky, is blown off by the wind; the riddle was at times a prominent literary form in the ancient and medieval world, 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"; the answers to the riddles are not preserved. "It is clear that we have here riddles from oral tradition that a teacher has put into a schoolbook." It is thought. "The Sanskrit term that most close
A paradox is a statement that, despite valid reasoning from true premises, leads to an apparently-self-contradictory or logically unacceptable conclusion. A paradox involves contradictory-yet-interrelated elements that exist 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, have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, are not yet resolved. Examples outside logic include the ship of Theseus from philosophy. Paradoxes can take the form of images or other media. For example, M. C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, staircases that appear to climb endlessly.
In common usage, the word "paradox" refers to statements that are 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, confusion between different levels of abstraction. 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". Another example of contradiction is if a man talking to a genie wishes that wishes couldn't come true; this contradicts itself because if the genie grants his wish, he did not grant his wish, if he refuses to grant his wish he did indeed grant his wish, therefore making it impossible either to grant or not grant his wish because his wish contradicts itself.
Vicious circularity, or infinite regress "This statement is false". Another example of vicious circularity is the following group of statements: "The following sentence is true." "The previous sentence is false."Other paradoxes involve false statements or half-truths and the resulting 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. Upon entering the surgery-suite, the surgeon says, "I can't operate on this boy. He's my son." The apparent paradox is caused by a hasty generalization, for if the surgeon is the boy's father, the statement cannot be true. The paradox is resolved. Paradoxes which are not based on a hidden error occur at the fringes of context or language, require extending the context or language in order to lose their paradoxical quality. Paradoxes that arise from intelligible uses of language are of interest to logicians and philosophers.
"This sentence is false" is an example of the well-known liar paradox: it is a sentence which cannot be interpreted as either true or false, because if it is known to be false it is known that it must be true, if it is known to be true it is known that it must be false. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory. Thought-experiments can yield interesting paradoxes; the grandfather paradox, for example, would arise if a time-traveler were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. This is a specific example of the more general observation of the butterfly effect, or that a time-traveller's interaction with the past—however slight—would entail making changes that would, in turn, change the future in which the time-travel was yet to occur, would thus change the circumstances of the time-travel itself.
A paradoxical conclusion arises from an inconsistent or inherently contradictory definition of the initial premise. In the case of that apparent paradox of a time-traveler killing his own grandfather, it is the inconsistency of defining the past to which he returns as being somehow different from the one which leads up to the future from which he begins his trip, but insisting that he must have come to that past from the same future as the one that it leads up to. W. V. Quine distinguished between three classes of paradoxes: A veridical paradox produces a result that appears absurd but is demonstrated to be true nonetheless, thus the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays if he had been born on a leap day. Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people; the Monty Hall paradox demonstrates that a decision which has an intuitive 50–50 chance is in fact biased towards making a decision which, given the intuitive concl