Counting is the process of determining the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once, until no unmarked elements are left; the related term enumeration refers to uniquely identifying the elements of a finite set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one. There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was used by ancient cultures to keep track of social and economic data such as number of group members, prey animals, property, or debts. Notched bones were found in the Border Caves in South Africa that may suggest that the concept of counting was known to humans as far back as 44,000 BCE; the development of counting led to the development of mathematical notation, numeral systems, writing.
Counting can occur in a variety of forms. Counting can be verbal; this is used to count objects that are present instead of counting a variety of things over time. Counting can be in the form of tally marks, making a mark for each number and counting all of the marks when done tallying; this is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting. Computers use base 2 counting. Counting can be in the form of finger counting when counting small numbers; this is used by children to facilitate counting and simple mathematical operations. Finger-counting uses unary notation, is thus limited to counting 10. Older finger counting used the four fingers and the three bones in each finger to count to the number twelve. Other hand-gesture systems are in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. By using finger binary, it is possible to keep a finger count up to 1023 = 210 − 1.
Various devices can be used to facilitate counting, such as hand tally counters and abacuses. Inclusive counting is encountered when dealing with time in the Romance languages. In exclusive counting languages such as English, when counting "8" days from Sunday, Monday will be day 1, Tuesday day 2, the following Monday will be the eighth day; when counting "inclusively," the Sunday will be day 1 and therefore the following Sunday will be the eighth day. For example, the French phrase for "fortnight" is quinzaine, similar words are present in Greek and Portuguese. In contrast, the English word "fortnight" itself derives from "a fourteen-night", as the archaic "sennight" does from "a seven-night". Names based on inclusive counting appear in other calendars as well: in the Roman calendar the nones is 8 days before the ides. Musical terminology uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc. and going up seven notes is an octave.
Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's first step into mathematics, constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback do not count, their languages do not have number words. Many children at just 2 years of age have some skill in reciting the count list, they can answer questions of ordinality for small numbers, e.g. "What comes after three?". They can be skilled at pointing to each object in a set and reciting the words one after another; this leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set. Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed. In the meantime, children learn. In mathematics, the essence of counting a set and finding a result n, is that it establishes a one-to-one correspondence of the set with the set of numbers.
A fundamental fact, which can be proved by mathematical induction, is that no bijection can exist between and unless n = m. This is the fundamental mathematical theorem. In a broader context, the theorem is an example of a theorem in the mathematical field of combinatorics—hence combinatorics is sometimes referred to as "the mathematics of counting." Many sets that arise in m
The Mughal Empire or Mogul Empire was an empire in the Indian subcontinent, founded in 1526. It was established and ruled by the Timurid dynasty, with Turco-Mongol Chagatai roots from Central Asia, claiming direct descent from both Genghis Khan and Timur, with significant Indian Rajput and Persian ancestry through marriage alliances; the dynasty was Indo-Persian in culture, combining Persianate culture with local Indian cultural influences visible in its court culture and administrative customs. The beginning of the empire is conventionally dated to the victory by its founder Babur over Ibrahim Lodi, the last ruler of the Delhi Sultanate, in the First Battle of Panipat. During the reign of Humayun, the successor of Babur, the empire was interrupted by the Sur Empire established by Sher Shah Suri; the "classic period" of the Mughal Empire began with the ascension of Akbar to the throne. Some Rajput kingdoms continued to pose a significant threat to the Mughal dominance of northwestern India, but most of them were subdued by Akbar.
All Mughal emperors were Muslims. The Mughal Empire did not try to intervene in native societies during most of its existence, rather co-opting and pacifying them through concilliatory administrative practices and a syncretic, inclusive ruling elite, leading to more systematic and uniform rule. Traditional and newly coherent social groups in northern and western India, such as the Marathas, the Rajputs, the Pashtuns, the Hindu Jats and the Sikhs, gained military and governing ambitions during Mughal rule which, through collaboration or adversity, gave them both recognition and military experience. Internal dissatisfaction arose due to the weakness of the empire's administrative and economic systems, leading to its break-up and declarations of independence of its former provinces by the Nawab of Bengal, the Nawab of Awadh, the Nizam of Hyderabad and other small states. In 1739, the Mughals were crushingly defeated in the Battle of Karnal by the forces of Nader Shah, the founder of the Afsharid dynasty in Persia, Delhi was sacked and looted, drastically accelerating their decline.
By the mid-18th century, the Marathas had routed Mughal armies and won over several Mughal provinces from the Punjab to Bengal. During the following century Mughal power had become limited, the last emperor, Bahadur Shah II, had authority over only the city of Shahjahanabad. Bahadur issued a firman supporting the Indian Rebellion of 1857. Consequent to the rebellion's defeat he was tried by the British East India Company for treason and exiled to Rangoon; the last remnants of the empire were formally taken over by the British, the British Parliament passed the Government of India Act 1858 to enable the Crown formally to displace the rights of the East India Company and assume direct control of India in the form of the new British Raj. At its height, the Mughal Empire stretched from Kabul, Afghanistan in the west to Arakan, Myanmar in the east, from Kashmir in the north to the Deccan Plateau in the south, extending over nearly all of the Indian subcontinent, it was the third largest empire in the Indian subcontinent, spanning four million square kilometers at its zenith, 122% of the size of the modern Republic of India.
The maximum expansion was reached during the reign of Aurangzeb, who ruled over more than 150 million subjects, nearly 25% of the world's population at the time. The Mughal Empire ushered in a period of proto-industrialization, around the 17th century, Mughal India became the world's largest economic and manufacturing power, responsible for 25% of global industrial output until the 18th century; the Mughal Empire is considered "India's last golden age" and one of the three Islamic Gunpowder Empires. The reign of Shah Jahan represented the height of Mughal architecture, with famous monuments such as the Taj Mahal, Moti Masjid, Red Fort, Jama Masjid and Lahore Fort being constructed during his reign. Contemporaries referred to the empire founded by Babur as the Timurid empire, which reflected the heritage of his dynasty, this was the term preferred by the Mughals themselves; the Mughal designation for their own dynasty was Gurkani. The use of Mughal derived from the Arabic and Persian corruption of Mongol, it emphasised the Mongol origins of the Timurid dynasty.
The term remains disputed by Indologists. Similar terms had been used to refer to the empire, including "Mogul" and "Moghul". Babur's ancestors were distinguished from the classical Mongols insofar as they were oriented towards Persian rather than Turco-Mongol culture. Another name for the empire was Hindustan, documented in the Ain-i-Akbari, and, described as the closest to an official name for the empire. In the west, the term "Mughal" was used for the emperor, by extension, the empire as a whole; the Mughal Empire was founded by Babur, a Central Asian ruler, descended from the Turco-Mongol conqueror Timur on his father's side and from Chagatai, the second son of the Mongol ruler Genghis Khan, on his mother's side. Ousted from his ancestral domains in C
A board game is a tabletop game that involves counters or pieces moved or placed on a pre-marked surface or "board", according to a set of rules. Some games are based on pure strategy. Games have a goal that a player aims to achieve. Early board games represented a battle between two armies, most modern board games are still based on defeating opponents in terms of counters, winning position, or accrual of points. There are many varieties of board games, their representation of real-life situations can range from having no inherent theme, like checkers, to having a specific theme and narrative, like Cluedo. Rules can range from the simple, like Tic-tac-toe, to those describing a game universe in great detail, like Dungeons & Dragons – although most of the latter are role-playing games where the board is secondary to the game, serving to help visualize the game scenario; the time required to learn to play or master a game varies from game to game, but is not correlated with the number or complexity of rules.
Board games have been played in societies throughout history. A number of important historical sites and documents shed light on early board games such as Jiroft civilization gameboards in Iran. Senet, found in Predynastic and First Dynasty burials of Egypt, c. 3500 BC and 3100 BC is the oldest board game known to have existed. Senet was pictured in a fresco found in Merknera's tomb. From predynastic Egypt is Mehen. Hounds and Jackals another ancient Egyptean board game appeared around 2000 BC; the first complete set of this game was discovered from a Theban tomb that dates to the 13th Dynasty. This game was popular in Mesopotamia and the Caucasus. Backgammon originated in ancient Persia over 5,000 years ago. Chess and Chaupar originated in India. Go and Liubo originated in China. Patolli originated in Mesoamerica played by the ancient Aztec and The Royal Game of Ur was found in the Royal Tombs of Ur, dating to Mesopotamia 4,600 years ago; the earliest known games list is the Buddha games list. In 17th and 18th century colonial America, the agrarian life of the country left little time for game playing though draughts and card games were not unknown.
The Pilgrims and Puritans of New England frowned on game playing and viewed dice as instruments of the devil. When the Governor William Bradford discovered a group of non-Puritans playing stool-ball, pitching the bar, pursuing other sports in the streets on Christmas Day, 1622, he confiscated their implements, reprimanded them, told them their devotion for the day should be confined to their homes. In Thoughts on Lotteries Thomas Jefferson wrote: Almost all these pursuits of chance produce something useful to society, but there are some which produce nothing, endanger the well-being of the individuals engaged in them or of others depending on them. Such are games with cards, billiards, etc, and although the pursuit of them is a matter of natural right, yet society, perceiving the irresistible bent of some of its members to pursue them, the ruin produced by them to the families depending on these individuals, consider it as a case of insanity, quoad hoc, step in to protect the family and the party himself, as in other cases of insanity, imbecility, etc. and suppress the pursuit altogether, the natural right of following it.
There are some other games of chance, useful on certain occasions, injurious only when carried beyond their useful bounds. Such are insurances, raffles, etc; these they do not take their regulation under their own discretion. The board game Traveller's Tour Through the United States and its sister game Traveller's Tour Through Europe were published by New York City bookseller F. & R. Lockwood in 1822 and today claims the distinction of being the first board game published in the United States; as the U. S. shifted from agrarian to urban living in the 19th century, greater leisure time and a rise in income became available to the middle class. The American home, once the center of economic production, became the locus of entertainment and education under the supervision of mothers. Children were encouraged to play board games that developed literacy skills and provided moral instruction; the earliest board games published in the United States were based upon Christian morality. The Mansion of Happiness, for example, sent players along a path of virtues and vices that led to the Mansion of Happiness.
The Game of Pope and Pagan, or The Siege of the Stronghold of Satan by the Christian Army pitted an image on its board of a Hindu woman committing suttee against missionaries landing on a foreign shore. The missionaries are cast in white as "the symbol of innocence and hope" while the pope and pagan are cast in black, the color of "gloom of error, and... grief at the daily loss of empire". Commercially produced board games in the mid-19th century were monochrome prints laboriously hand-colored by teams of low-paid young factory women. Advances in paper making and printmaking during the period enabled the commercial production of inexpensive board games; the most significant advance was the development of chromolithography, a technological achievement that made bold, richly colored images available at affordable prices. Games cost as little as US$.25 for a small boxed card game to $3.00 for more elaborate games. American Protestants believed a virtuous life led to success, but the belief was challenged mid-century when the country embraced materialism and c
Dice are small throwable objects that can rest in multiple positions, used for generating random numbers. Dice are suitable as gambling devices for games like craps and are used in non-gambling tabletop games. A traditional die is a cube, with each of its six faces showing a different number of dots from one to six; when thrown or rolled, the die comes to rest showing on its upper surface a random integer from one to six, each value being likely. A variety of similar devices are described as dice, they may be used to produce results other than one through six. Loaded and crooked dice are designed to favor some results over others for purposes of cheating or amusement. A dice tray, a tray used to contain thrown dice, is sometimes used for gambling or board games, in particular to allow dice throws which do not interfere with other game pieces. Dice have been used since before recorded history, it is uncertain where they originated; the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800–2500 BC.
Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin. The Egyptian game of Senet was played with dice. Senet was played before 3000 BC and up to the 2nd century AD, it was a racing game, but there is no scholarly consensus on the rules of Senet. Dicing is mentioned as an Indian game in the Rigveda and the early Buddhist games list. There are several biblical references to "casting lots", as in Psalm 22, indicating that dicing was commonplace when the psalm was composed, it is theorized that dice developed from the practice of fortunetelling with the talus of hoofed animals, colloquially known as "knucklebones", but knucklebones is not the oldest divination technique that incorporates randomness. Knucklebones was a game of skill played by children. Although gambling was illegal, many Romans were passionate gamblers who enjoyed dicing, known as aleam ludere. Dicing was a popular pastime of emperors. Letters by Augustus to Tacitus and his daughter recount his hobby of dicing.
There were two sizes of Roman dice. Tali were large dice inscribed with one, three and six on four sides. Tesserae were smaller dice with sides numbered from one to six. Twenty-sided dice date back to the 2nd century AD and from Ptolemaic Egypt as early as the 2nd century BC. Dominoes and playing cards originated in China as developments from dice; the transition from dice to playing cards occurred in China around the Tang dynasty, coincides with the technological transition from rolls of manuscripts to block printed books. In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. Ban-sugoroku is similar to backgammon and dates to the Heian period, while e-sugoroku is a racing game. Dice are thrown onto a surface either from a container designed for this; the face of the die, uppermost when it comes to rest provides the value of the throw. One typical dice game today is craps, where two dice are thrown and wagers are made on the total value of the two dice.
Dice are used to randomize moves in board games by deciding the distance through which a piece will move along the board. The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics. A die roll is made random by uncertainty in minor factors such as tiny movements in the thrower's hand. To mitigate concerns that the pips on the faces of certain styles of dice cause a small bias, casinos use precision dice with flush markings. Common dice are small cubes most 1.6 cm across, whose faces are numbered from one to six by patterns of round dots called pips. Opposite sides of a modern die traditionally add up to seven, implying that the 1, 2 and 3 faces share a vertex; the faces of a die may be placed counterclockwise about this vertex. If the 1, 2 and 3 faces run counterclockwise, the die is called "right-handed", if those faces run clockwise, the die is called "left-handed". Western dice are right-handed, Chinese dice are left-handed; the pips on dice are arranged in specific patterns.
Asian style dice bear similar patterns to Western ones, but the pips are closer to the center of the face. One possible explanation is. In some older sets, the "one" pip is a colorless depression. Non-precision dice are manufactured via the plastic injection molding process; the pips or numbers on the die are a part of the mold. The coloring for numbering is achieved by submerging the die in paint, allowed to dry; the die is polished via a tumble finishing process similar to rock polishing. The abrasive agent scrapes off all of the paint except for the indents of the numbering. A finer abrasive is used to polish the die; this process creates the smoother, rounded edges on the dice. Precision casino dice may have a polished or sand finish, making them transparent or translucent res
The Indian subcontinent known as the Asian subcontinent and Indo subcontinent, is a southern region and peninsula of Asia situated on the Indian Plate and projecting southwards into the Indian Ocean from the Himalayas. Geologically, the Indian subcontinent is related to the land mass that rifted from Gondwana and merged with the Eurasian plate nearly 55 million years ago. Geographically, it is the peninsular region in south-central Asia delineated by the Himalayas in the north, the Hindu Kush in the west, the Arakanese in the east. Politically, the Indian subcontinent includes Bangladesh, India, Nepal and Sri Lanka. Sometimes, the geographical term'Indian subcontinent' is used interchangeably with'South Asia', although that last term is used as a political term and is used to include Afghanistan. Which countries should be included in either of these remains the subject of debate. According to Oxford English Dictionary, the term "subcontinent" signifies a "subdivision of a continent which has a distinct geographical, political, or cultural identity" and a "large land mass somewhat smaller than a continent".
It is first attested in 1845 to refer to the North and South Americas, before they were regarded as separate continents. Its use to refer to the Indian subcontinent is seen from the early twentieth century, it was convenient for referring to the region comprising both British India and the princely states under British Paramountcy. The term Indian subcontinent has a geological significance. Similar to various continents, it was a part of the supercontinent of Gondwana. A series of tectonic splits caused formation of various basins, each drifting in various directions; the geological region called "Greater India" once included Madagascar, Seychelles and Austrolasia along with the Indian subcontinent basin. As a geological term, Indian subcontinent has meant that region formed from the collision of the Indian basin with Eurasia nearly 55 million years ago, towards the end of Paleocene; the geographical region has simply been known as "India". Other related terms are South Asia, and the terms "Indian subcontinent" and "South Asia" are sometimes used interchangeably.
There is no globally accepted definition on which countries are a part of South Asia or the Indian subcontinent. The less common term "South Asian subcontinent" has seen occasional use since the 1970s. Geologically, the Indian subcontinent was first a part of so-called "Greater India", a region of Gondwana that drifted away from East Africa about 160 million years ago, around the Middle Jurassic period; the region experienced high volcanic activity and plate subdivisions, creating Madagascar, Antarctica and the Indian subcontinent basin. The Indian subcontinent drifted northeastwards, colliding with the Eurasian plate nearly 55 million years ago, towards the end of Paleocene; this geological region includes Bangladesh, India, Nepal and Sri Lanka. The zone where the Eurasian and Indian subcontinent plates meet remains one of the geologically active areas, prone to major earthquakes; the English term "subcontinent" continues to refer to the Indian subcontinent. Physiographically, it is a peninsular region in south-central Asia delineated by the Himalayas in the north, the Hindu Kush in the west, the Arakanese in the east.
It extends southward into the Indian Ocean with the Arabian Sea to the southwest and the Bay of Bengal to the southeast. Most of this region rests on the Indian Plate and is isolated from the rest of Asia by large mountain barriers. Using the more expansive definition – counting India, Bangladesh, Sri Lanka, Nepal and Maldives as the constituent countries – the Indian subcontinent covers about 4.4 million km2, 10% of the Asian continent or 3.3% of the world's land surface area. Overall, it is home to a vast array of peoples; the Indian subcontinent is a natural physical landmass in South Asia, geologically the dry-land portion of the Indian Plate, isolated from the rest of Eurasia. Given the difficulty of passage through the Himalayas, the sociocultural and political interaction of the Indian subcontinent has been through the valleys of Afghanistan in its northwest, the valleys of Manipur in its east, by maritime routes. More difficult but important interaction has occurred through passages pioneered by the Tibetans.
These routes and interactions have led to the spread of Buddhism out of the Indian subcontinent into other parts of Asia. And the Islamic expansion arrived into the Indian subcontinent in two ways, through Afghanistan on land and to Indian coast through the maritime routes on the Arabian Sea. Whether called the Indian subcontinent or South Asia, the definition of the geographical extent of this region varies. Geopolitically, it had formed the whole territory of Greater India. In terms of modern geopolitical boundaries, the Indian subcontinent comprises the Republic of India, Bangladesh, Bhutan, besides, by convention, the island nation of Sri Lanka and other islands of the Indian Ocean, such as the Maldives; the term "Indian continent" is first introduced in the early 20th century, when most of the territory was part of British India. The Hindu Kush, centered on eastern Afghanistan, is the boundary connecting the Indian subcontinent with Central Asia to the northwest, the Persian Plateau to the west.
The socio-religious history of Afghanistan are related to the Turkish-influenced Central Asia and no
Trouble (board game)
Trouble is a board game in which players compete to be the first to send four pieces all the way around a board. Pieces are moved according to the roll of a die. Trouble was developed by the Kohner Brothers and manufactured by Irwin Toy Ltd. by Milton Bradley. The game was launched in the United States in 1965 and is based on the English game Ludo and the German game Mensch ärgere dich nicht, both derived from the classic Indian game Pachisi; the Hasbro game Sorry! has a similar game mechanic but uses cards instead of dice. The classic version is now marketed by Winning Moves. A similar game called Headache was produced by the Milton Bradley Company. A world championship of "Frustration", as it's known in the UK, is held biannually in the Isle of Wight over two days, the morphing of a number of smaller regional leagues and tournaments played over the last two decades. In the World Championship the first year's championship is decided on 31 December and the second on 1 January. Substantial cash prizes are awarded to the winner with 2012 world champion being Elizabeth O'Brien from Woolton near Liverpool.
O'Brien retained her world crown on 31 December 2013, giving her three consecutive victories and a substantial cash prize. The current 2014 world champion is artisan Andrew Rimmer. Celebrity players of the cult game include renowned travel author Tim Moore and actor-presenter Simon O'Brien, husband of Elizabeth. Moore is the current holder of the Elite Premier World League first played in southern France in summer 2013, which presented a significant cash prize for the winner; the second EPWL – an invitation tournament – was played in the Spanish region of Valencia in June 2014. Participants came from France and the UK to participate; the first EPWL as won by a player known as Spanish Ports who defeated Moore and other unspecified players. The most recent world championship was held on the Isle of Wight on 31 December 2016; the game in the UK has developed internal jargon such as "co" for "cocked dice". The EPWL and World Championship organising structure are outlining plans to connect with Finnish Kimble players to start-up a pan-European tournament framework.
Players can send opponents' pieces back to the start by landing on them. Players may not touch the other player's piece, otherwise the piece's owner must redo the pop/roll. Teaming is not allowed in the game. Pieces are protected from capture after arriving in the final four slots. Unlike more complex race games, counters cannot be maneuvered to block opponents' moves; the most notable feature of Trouble is the "Pop-O-Matic" die container. This device is a clear plastic hemisphere containing the die, placed over a flexible sheet. Players roll the die by pressing down on the bubble, which flexes the sheet and causes the die to tumble upon its rebound; the Pop-O-Matic container produces a popping sound when it is used, prevents the die from being lost. It allows for quick die rolls, players' turns can be performed in rapid succession; the die is imprinted with Arabic numerals rather than the traditional circular pips. Players may move pieces out of their start only when the die lands on 6. Getting a 6 at any point in the board game allows the player to take another turn if the player cannot move any of their pieces.
They can move a new piece out if they have another piece in play, can do the same if another player's piece is occupying their "start" space, but cannot do so when one of their own pieces is occupying their "start" space. If the die in the "Pop-O-Matic" container has not landed on a number the player who popped it can tap the "Pop-O-Matic", but may not re-pop while the die is in limbo. Trouble: Fun on the Run edition. Trouble: 2013 edition. Trouble: Cars 2 edition. Trouble: Star Wars: The Clone Wars edition. Trouble: Despicable Me edition. Trouble: Frozen edition. Trouble: SpongeBob SquarePants edition. Trouble: Toy Story 3 edition. Trouble: Cars 3 edition. Trouble and Double Trouble at BoardGameGeek
Abstract strategy game
An abstract strategy game is a strategy game in which the theme is not important to the experience of playing. Many of the world's classic board games, including chess, Go, checkers and draughts, shogi, Nine Men's Morris, most mancala variants, fit into this category. Play is sometimes said to resemble a series of puzzles; as J. Mark Thompson wrote in his article "Defining the Abstract": There is an intimate relationship between such games and puzzles: every board position presents the player with the puzzle, What is the best move?, which in theory could be solved by logic alone. A good abstract game can therefore be thought of as a "family" of interesting logic puzzles, the play consists of each player posing such a puzzle to the other. Good players are the ones. Many abstract strategy games happen to be "combinatorial". Combinatorial games have no randomizers such as dice, no simultaneous movement, nor hidden information; some games that do have these elements are sometimes classified as abstract strategy games.
A smaller category of abstract strategy games manages to incorporate hidden information without using any random elements. Traditional abstract strategy games are treated as a separate game category, hence the term'abstract games' is used for competitions that exclude them and can be thought of as referring to modern abstract strategy games. Two examples are the IAGO World Tour and the Abstract Games World Championship held annually since 2008 as part of the Mind Sports Olympiad; some abstract strategy games have multiple starting positions of which it is required that one be randomly determined. For a game to be one of skill, a starting position needs to be chosen by impartial means; some games, such as Arimaa and DVONN, have the players build the starting position in a separate initial phase which itself conforms to combinatorial game principles. Most players, would consider that although one is starting each game from a different position, the game itself contains no luck element. Indeed, Bobby Fischer promoted randomization of the starting position in chess in order to increase player dependence on thinking at the board.
Analysis of "pure" abstract strategy games is the subject of combinatorial game theory. Abstract strategy games with hidden information, bluffing, or simultaneous move elements are better served by Von Neumann–Morgenstern game theory, while those with a component of luck may require probability theory incorporated into either of the above; as for the qualitative aspects, ranking abstract strategy games according to their interest, complexity, or strategy levels is a daunting task and subject to extreme subjectivity. In terms of measuring how finite a mathematical field each of the three top contenders represents, it is estimated that checkers has a game-tree complexity of 1031 possible positions, whereas chess has 10123; this suggests that computer programs, through brute force calculation alone, should be able to surpass human players' abilities. As for Go, the possible legal game positions range in the magnitude of 10170; the Mind Sports Olympiad first held the Abstract Games World Championship in 2008 to try to find the best abstract strategy games all-rounder.
The MSO event saw a change in format in 2011 restricting the competition to players' five best events, was renamed to the Modern Abstract Games World Championship. It was again won by David Pearce. 2008: David M. Pearce 2009: David M. Pearce 2010: David M. Pearce 2011: David M. Pearce 2012: Andres Kuusk 2013: Andres Kuusk Connection games Game complexity List of abstract strategy games List of world championships in mind sports Mind Sports Olympiad World Mind Sports Games The University of Alberta Games Group David Eppstein's CGT page