Luck, Lottery, Combinatorial , mathematics, Factorial , Numeral system, List of topics

Contents Articles Luck

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Lottery

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Combinatorial number system

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Gaming mathematics

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Factorial number system

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Numeral system

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List of numeral system topics

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References Article Sources and Contributors

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Image Sources, Licenses and Contributors

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Article Licenses License

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Luck

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Luck Luck or fortuity is good or bad fortune in life caused by accident

or chance, and attributed by some to reasons of faith or superstition, which happens beyond a person's control. [1] [2] [3] Luck is pervasive in common speech.[4] Typical use includes "Good Luck!" to wish a blessing on someone, or describing a misfortune, as in "it was just bad luck." There are many expressions and quotes about Luck. [5] [6] Cultural views of luck vary from perceiving luck as a matter of  random chance to attributing to luck explanations of faith or superstition. For example, the Romans believed in i n the embodiment [7] of luck as the Goddess Fortuna, while the atheist and philosopher Daniel Dennett believes that "luck is mere luck" rather than a property of a person or thing. [8]

A four-leaf clover is often considered to bestow good luck.

Lucky Symbols have widespread global appeal and are represented by human, animal, botanical and inanimate objects.

Definition Luck is a way of understanding a personal chance event. Luck has three aspects [9] [10] which make it distinct from chance or probability. [11]

• Luck Luck is good good or bad. bad.[12] • Luck Luck is is by by acci acciden dentt or or chan chance. ce.[13] • Luck Luck app applie liess to a per perso son. n. Some examples of Luck: • You win win repeated repeatedly ly at gambling, gambling, against against significant significant odds. • You correctl correctlyy guess an answer answer in a quiz quiz which which you you didn't didn't know.

1926 US advertisement for lucky jewellery . "Why Be Unlucky?".

• Your car car breaking breaking down could could be bad bad luck, luck, if it was by chance chance and against against the odds. odds.

Interpretations of luck Luck is interpreted and understood in many different ways.

Luck as lack of control Luck refers to that which happens to a person beyond that person's control. This view incorporates phenomena that are chance happenings, a person's place of birth for example, but where there is no uncertainty involved, or where the uncertainty is irrelevant. Within this framework one can differentiate between three different types of luck: 1. Constitutional luck, that is, luck with factors that cannot be changed. changed. Place of birth and genetic constitution are typical examples. 2. Circumstantial luck - with factors that are haphazardly haphazardly brought on. Accidents and epidemics are typical typical examples. 3. Ignorance luck, that is, luck with factors one does not know about. Examples can be identified only in hindsight.

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Luck as a fallacy Another view holds that "luck is probability taken personally." A rationalist approach to luck includes the application of the rules of probability, and an avoidance of unscientific beliefs. The rationalist feels the belief in luck is a result of poor reasoning or wishful thinking. To a rationalist, a believer in luck who asserts that something has influenced his or her luck commits the "post hoc ergo propter hoc" logical fallacy: that because two events are connected sequentially, they are connected causally as well. In general:

 A happens (luck-attracting event or action) and then  B happens; Therefore, A influenced B. In the rationalist perspective, probability is only affected by confirmed causal connections. The gambler's fallacy and inverse gambler's fallacy both explain some reasoning problems in common beliefs in luck. They involve denying the unpredictability of random events: "I haven't rolled a seven all week, so I'll definitely roll one tonight". Luck is merely an expression noting an extended period of noted outcomes, completely consistent with random walk probability theory. Wishing one "good luck" will not cause such an extended period, but it expresses positive feelings toward the one —not necessarily wholly undesirable. It cannot be shown that luck actually exists, hence luck is nothing more than a word used by one in a self delusional assumption of understanding events of which one is informed or which one witnesses. As such, it is a word which superstitious people use to simultaneously presume to have insight into events and, paradoxically, to cease efforts to understand the causes and effects of those same events.

Luck as an essence There is also a series of spiritual, or supernatural beliefs regarding fortune. These beliefs vary widely from one to another, but most agree that luck can be influenced through spiritual means by performing certain rituals or by avoiding certain circumstances. One such activity is prayer, a religious practice in which this belief is particularly strong. Many cultures and religions worldwide place a strong emphasis on a person's ability to influence their fortune by ritualistic means, sometimes involving sacrifice, omens or spells. Others associate luck with a strong sense of superstition, that is, a belief that certain taboo or blessed actions will influence how fortune favors them for the future. Luck can also be a belief in an organization of fortunate and unfortunate events. Luck is a form of superstition which is interpreted differently by different individuals. Famous Swiss psychiatrist, Carl Jung, who founded analytical psychology, coined the term "synchronicity", which he described as "a meaningful coincidence". Christianity and Islam believe in the will of a supreme being rather than luck as the primary influence in future events. The degrees of this Divine Providence vary greatly from one person to another; however, most acknowledge providence as at least a partial, if not complete influence on luck. Christianity, in its early development, accommodated many traditional practices which at different times, accepted omens and practiced forms of ritual sacrifice in order to divine the will of their supreme being or to influence divine favoritism. The concept of "Divine Grace" as it is described by believers closely resembles what is referred to as "luck" by others.

Luck Mesoamerican religions, such as the Aztecs, Mayans and Incas, had particularly strong beliefs regarding the relationship between rituals and luck. In these cultures, human sacrifice (both of willing volunteers and captured enemies) was seen as a way to please the gods and earn favor for the city offering the sacrifice. The Mayans also believed in blood offerings, where men or women wanting to earn favor with the gods, to bring about good luck, would cut themselves and bleed on the gods' altar. Many traditional African practices, such as voodoo and hoodoo, have a strong belief in superstition. Some of these religions include a belief that third parties can influence an individual's luck. Shamans and witches are both respected and feared, based on their ability to cause good or bad fortune for those in villages near them.

Luck as a self-fulfilling prophecy Some encourage the belief in luck as a false idea, but which may produce positive thinking, and alter one's responses for the better. Others, like Jean-Paul Sartre and Sigmund Freud, feel a belief in luck has more to do with a locus of  control for events in one's life, and the subsequent escape from personal responsibility. According to this theory, one who ascribes their travails to "bad luck" will be found upon close examination to be living risky lifestyles. In personality psychology, people reliably differ from each other depending on four key aspects: beliefs in luck, rejection of luck, being lucky, and being unlucky. [14] People who believe in good luck are more optimistic, more satisfied with their lives, and have better moods.[14] If "good" and "bad" events occur at random to everyone, believers in good luck will experience a net gain in their fortunes, and vice versa for believers in bad luck. This is clearly likely to be self-reinforcing. Thus, a belief in good luck may actually be an adaptive meme.

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Luck

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Social aspects of luck Luck is an important factor in many aspects of society.

Games A Game may depend on luck rather than skill or effort. For example, Chess does not involve any random factors such as throwing dice, while Dominoes has the "luck of the draw" when selecting tiles.

Lotteries Many countries have a national lottery. Individual views of the chance of winning, and what it might mean to win, are largely expressed by statements about luck. For example, the winner was "just lucky" meaning they contributed no skill or effort.

Means of resolving issues "Leaving it to chance" is a way of resolving issues. For example, flipping a coin at the start of a sporting event may determine who goes first.

Numerology Most cultures consider some numbers to be lucky or unlucky. This is found to be particularly strong in Asian cultures, where the obtaining of "lucky" telephone numbers, automobile license plate numbers, and household addresses are actively sought, sometimes at great monetary expense. Numerology, as it relates to luck, is closer to an art than to a science, yet numerologists, astrologists or psychics may disagree. It is interrelated to astrology, and to some A National Lottery "play here!" sign outside a degree to parapsychology and spirituality and is based on newsagents on the Euston Road, London. converting virtually anything material into a pure number, using that number in an attempt to detect something meaningful about reality, and trying to predict or calculate the future based on lucky numbers. Numerology is folkloric by nature and started when humans first learned to count. Through human history it was, and still is, practiced by many cultures of the world from traditional fortune-telling to on-line psychic reading.

Luck in religion and mythology Buddhism Gautama Buddha, the founder of Buddhism, taught his followers not to believe in luck. The view which was taught by Gautama Buddha states that all things which happen must have a cause, either material or spiritual, and do not occur due to luck, chance or fate. The idea of moral causality, karma (Pali: kamma), is central in Buddhism. In the Sutta Nipata ,the Buddha is recorded as having said the following about luck: Whereas some religious men, while living of food provided by the faithful make their living by such low arts, such wrong means of livelihood as palmistry, divining by signs, interpreting dreams... bringing good or bad luck... invoking the goodness of luck... picking the lucky site for a building, the monk Gautama refrains from such low arts, such wrong means of livelihood. D.I, 9-12 [15]

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Nonetheless, belief in luck is overwhelmingly prevalent in many predominantly Buddhist Asian countries. In Thailand, for example, Buddhists may wear verses (takrut) or lucky amulets which have been blessed by monks for protection against physical and spiritual harm. [16]

Japanese mythology As represented by the Seven Lucky Gods, namely Hotei, Jurōjin, Fukurokuju, Bishamonten, Benzaiten, Daikokuten and Ebisu

Hinduism Lakshmi A Hindu Devi (English: Divinity) of Money & Fortune. It is said that by proper worship, with a meticulous prayer procedure (Sanskrit: Shri Lakshmi Sahasranam Pujan Vidhi) the blessings of this powerful deity may be obtained. Lakshmi Parayan (Prayer) is performed in most Hindu homes on the day of Diwali or the festival of lights.

Judaism and Christianity •  But you who forsake Yahweh, who forget my holy mountain, who prepare a table for Fortune, and who fill up mixed wine to Destiny (Isaiah 65:11 [17] - The bearing that this has on beliefs concerning luck is a matter of  controversy) • The lot is cast into the lap, but its every decision is from the Lord  (Book of Proverbs 16:33 NIV) •  I have seen something else under the sun: The race is not to the swift or the battle to the strong, nor does food  come to the wise or wealth to the brilliant or favor to the learned; but time and chance happen to them all. (Ecclesiastes 9:11 NIV)

Islam There is no concept of Luck in Islam [18] other than actions pre-determined by God(Allah) and that God alone has power over all things (Divine Decree). It is stated in the Qur'an (Sura: Adh-Dhariyat ( The Wind that Scatter ) verse:22) that one’s sustenance is pre-determined in heaven when the Lord says: “And in the heaven is your provision and that which ye are promised. ” However, one should supplicate towards God to better one's life rather than hold faith in un-Islamic acts such as using "lucky charms".

Roman Catholic Church The Catholic Church excludes chance or luck as an explanation for creation,

[19]

[20]

Wicca Many Wiccans believe in luck, and use spells, ritual and other forms of magic in an attempt to influence their own luck and the luck of others.

Sikhism Sikhism founded by Guru Nanak Dev in India. 1. No one can, can, by any any way, get get grapes by by the seed seed of acacia. acacia. 2. Luck is but but lack of self self confidenc confidencee and fruit fruit of idleness. idleness.

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See also • • • • •

Chance Fate Pro Probability Ser Serendipity Luc Lucky Symb Symbol olss

External links • Luck, Luck, Destin Destiny, y, Fate, Fate, Karm Karma, a, or Self Self-Ma -Made? de? [21] with psychologist Richard Wiseman • "Lucky "Lucky": ": Docume Documenta ntary ry with with Richard Richard Wisem Wiseman an [22] transcript with link to 10 minute video.

References [1] http://w http://wordn ordnetw etweb. eb.princeton. princeton.edu/perl/webwn?s=luck edu/perl/webwn?s=luck "an unknown and unpredictable phenomenon that causes an event to result one way rather than another" [2] http://d http://dict ictiona ionary. ry.reference. reference.com/browse/luck com/browse/luck "the force that seems to operate for good or ill in a person's life, as in shaping circumstances, events, or opportunities" [3] http http:// ://en en..wiktionary.org/wiki/luck wiktionary.org/wiki/luck [4] The 3000 Most Commonly Commonly Used Used Words in the United United States States (http://www. (http://www.paulnoll. paulnoll.com/Books/Clear-English/words-23-24-hundred. com/Books/Clear-English/words-23-24-hundred. html) Luck is listed at 2361. [5] Wikiqu Wikiquote ote:: Luck [6] http http:// ://ww www. w.buzzle. buzzle.com/articles/twenty-good-luck-quotes. com/articles/twenty-good-luck-quotes.html html [7] http http:// ://ww www. w.thaliatook. thaliatook.com/OGOD/fortuna. com/OGOD/fortuna.html html [8] Elbow Elbow Room Room (http://b (http://books ooks..google.com/books?id=6SPBOq1BCf0C&pg=PA92) by Daniel Clement Dennett, Page 92. "We know it would be superstitious to believe that "there actually is such a thing as luck" - something a rabbits foot might bring - but we nevertheless think there is an unsuperstitious and unmisleading way of characterizing events and properties as merely lucky." [9] Luck: the brilliant randomness of everyday life (http://books.google. com/books?id=tSeYae1nrwcC) Page 32. "Luck accordingly involves three things: (1) a beneficiary or maleficiary, (2) a development that is benign (positive) or malign (negative) from the stand point of the interests of the affected individual, and that, moreover, (3) is fortuitous (unexpected, chancy, unforeseeable.)" [10] CHANCE CHANCE News 4.15 (http://www.da (http://www.dartmouth rtmouth..edu/~chance/chance_news/recent_news/chance_news_4. 15.html) 15. html) ...the definition in the Oxford English dictionary: "the fortuitous happening of an event favorable or unfavorable to the interest of a person" [11] Luck: the brilliant brilliant randomness of everyday life (http://books.google.com/ books?id=tSeYae1nrwcC) Page 28. "Luck is a matter of having something good or bad happen that lies outside the horizon of effective foreseeability." [12] Luck: the brilliant randomness of everyday life (http://books.google. com/books?id=tSeYae1nrwcC) Page 32. "Luck thus always incorporates a normative element of good or bad: someone must be affected positively or negatively by an event before its realization can properly be called lucky." [13] Luck: the brilliant brilliant randomness of everyday life (http://books.google.com/ books?id=tSeYae1nrwcC) Page 32. ..."that as a far as the affected person is concerned, the outcome came about "by accident." " [14] Maltby, J., Day, L., Gill, P., Colley, A., Wood, A. M. (2008). Beliefs around luck: Confirming the empirical conceptualization of beliefs around luck and the development of the Darke and Freedman beliefs around luck scale (http://personalpages. (http://personalpages.manchester. manchester.ac. ac.uk/staff/alex. uk/staff/alex. wood/Luck.pdf) wood/Luck. pdf) Personality and Individual Differences, 45 , 655-660. [15] [15] (htt (http: p://w //www ww..buddhanet.net/e-learning/qanda09. buddhanet.net/e-learning/qanda09.htm) htm) [16] [16] (htt (http: p://w //www ww..thailandlife.com/amulet. thailandlife.com/amulet.html) html) [17] http://e http://ebibl bible. e.org/web/Isaiah.htm#C65V11 org/web/Isaiah.htm#C65V11 [18] [18] http http:// ://ww www. w.islamweb. islamweb.net/ver2/Fatwa/ShowFatwa. net/ver2/Fatwa/ShowFatwa. php?lang=E&Id=91036&Option=FatwaId [19] God creates creates by wisdom wisdom and and love #295 #295 (http://www. (http://www.vatican. vatican.va/archive/catechism/p1s2c1p4. va/archive/catechism/p1s2c1p4. htm) "We believe that God created the world according to his wisdom. It is not the product of any necessity whatever, nor of blind fate or chance." [20] RESPECT RESPECT FOR PERSONS PERSONS AND AND THEIR GOODS GOODS #2413 (http://www. (http://www.vatican.va/archive/ccc_css/archive/catechism/p3s2c2a7. vatican.va/archive/ccc_css/archive/catechism/p3s2c2a7. htm) "Games of chance (card games, etc.) or wagers are not in themselves contrary to justice." [21] [21] http http:// ://ww www. w.fastcompany. fastcompany.com/magazine/72/realitycheck.html com/magazine/72/realitycheck.html [22] [22] http http:// ://ww www. w.abc. abc.net. net.au/catalyst/stories/2220191.htm au/catalyst/stories/2220191.htm

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Lottery A lottery is a form of gambling which involves the drawing of lots for a prize. The word stems from the Dutch word loterij, which is derived from the noun lot meaning fate or destiny. Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments. At the beginning of the 20th century, most forms of gambling, including lotteries and sweepstakes, were illegal in many countries, including the U.S.A. and most of Europe. This remained so until after World War II. In the 1960s casinos and lotteries began to appear throughout the world as a means to raise revenue in addition to taxes. Lotteries come in many formats. The prize can be a fixed amount of  cash or goods. In this format there is risk to the organizer if insufficient tickets are sold. More commonly the prize fund will be a fixed percentage of the receipts. A popular form of this is the "50 – 50" 50" draw where the organizers promise that the prize will be 50% of the revenue. Many recent lotteries allow purchasers to select the numbers on the lottery ticket resulting in the possibility of multiple winners. The purchase of lottery tickets is, from the perspective of classical economics, irrational. However, in addition to the chance of winning, the ticket may enable some purchasers to experience a thrill and to indulge in a fantasy of becoming wealthy. If the entertainment value (or other non-monetary value) obtained by playing is high enough for a given individual, then the purchase of a lottery ticket could represent a gain in overall utility. In such a case, the monetary loss would be outweighed by the non-monetary gain, thus making the purchase a rational decision for that individual.

National Lottery building located on Paseo de la Reforma in Mexico City.

Early history The first recorded signs of a lottery are Keno slips from the Chinese Han Dynasty between 205 and 187 B.C. These lotteries are believed to have helped to finance major government projects like the Great Wall of China. From the Chinese "The Book of Songs" (second millennium B.C.) comes a reference to a game of chance as "the drawing of  wood", which in context appears to describe the drawing of lots. From the Celtic era, the Cornish words "teulet pren" translates into "to throw wood" and means "to draw lots". The Iliad of Homer refers to lots being placed into Agamemnon's helmet to determine who would fight Hector. The first known European lotteries were held during the Roman Empire, mainly as an amusement at dinner parties. Each guest would receive a ticket, and prizes would often consist of fancy items such as dinnerware. Every ticket holder would be assured of winning something. This type of lottery, however, was no more than the distribution of  gifts by wealthy noblemen during the Saturnalian revelries. The earliest records of a lottery offering tickets for sale is the lottery organized by Roman Emperor Augustus Caesar. The funds were for repairs in the City of Rome, and the winners were given prizes in the form of articles of unequal value. The first recorded lotteries to offer tickets for sale with prizes in the form of money were held in the Low Countries in the 15th century. Various towns held public lotteries to raise money for town fortifications, and to help the poor. The town records of Ghent, Utrecht, and Bruges indicate that lotteries may be even older. A record dated May 9, 1445 at L'Ecluse refers to raising funds to build walls and town fortifications, with a lottery of 4,304 tickets and total prize money of 1737 florins.[1] In the 17th century it was quite usual in the Netherlands to organize lotteries to collect money for the poor or in order to raise funds for al kinds of public usages. The lotteries proved very popular

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and were hailed as a painless form of taxation. The Dutch state-owned Staatsloterij is the oldest running lottery.

England, 1566 –1826 Although the English probably first experimented with raffles and similar games of chance, the first recorded official lottery was chartered by Queen Elizabeth I, in the year 1566, and was drawn in 1569. This lottery was designed to raise money for the "reparation of the havens and strength of the Realme, and towardes such other publique good workes." Each ticket holder won a prize, and the total value of the prizes equalled the money raised. Prizes were in the form of silver plate and other valuable commodities. The lottery was promoted by scrolls posted throughout the country showing sketches of the prizes. [2] Thus, the lottery money received was an interest free loan to the government during the three years that the tickets ('without any Blankes') were sold. In later years, the government sold the lottery ticket rights to brokers, who in turn hired agents and runners to sell them. These brokers eventually became the modern day stockbrokers for various commercial ventures. Most people could not afford the entire cost of a lottery ticket, so the brokers would sell shares in a ticket; this resulted in tickets being issued with a notation such as "Sixteenth" or "Third Class."

English Lottery 1566 Scroll.

Many private lotteries were held, including raising money for The Virginia Company of London to support its settlement in America at Jamestown. The English State Lottery ran from 1694 until 1826. Thus, the English lotteries ran for over 250 years, until the government, under constant pressure from the opposition in parliament, declared a final lottery in English State Lottery Ticket 1814 issued by broker Swift & Co. 1826. This lottery was held up to ridicule by contemporary commentators as "the last struggle of the speculators on public credulity for popularity to their last dying lottery."

Early America, 1612 –1900 An English lottery, authorized by King James I in 1612, granted the Virginia Company of London the right to raise money to help establish settlers in the first permanent English colony at Jamestown, Virginia. Lotteries in colonial America played a significant part Ticket from an 1814 lottery to raise money for Queen's College, New in the financing of both private and public ventures. It Jersey. has been recorded that more than 200 lotteries were sanctioned between 1744 and 1776, and played a major role in financing roads, libraries, churches, colleges, canals, bridges, etc. [3] In the 1740s, the foundation of Princeton and Columbia Universities was financed by lotteries, as was the University of Pennsylvania by the Academy Lottery in 1755.

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During the French and Indian Wars, several colonies used lotteries to help finance fortifications and their local militia. In May 1758, the State of Massachusetts raised money with a lottery for the "Expedition against Canada." Benjamin Franklin organized a lottery to raise money to purchase cannon for the defense of Philadelphia. Several of these lotteries offered prizes in the form of  "Pieces of Eight." George Washington's Mountain Road Lottery in 1768 was unsuccessful. However, these rare lottery tickets bearing George Washington's signature have become collectors' items which sold for Massachusetts Lottery Ticket 1758 French & Indian Wars about $15,000 in 2007. Later, in 1769, Washington was a manager for Col. Bernard Moore's "Slave Lottery", which advertised land and slaves as prizes in the Virginia Gazette. At the outset of the Revolutionary War, the Continental Congress used lotteries to raise money to support the Colonial Army. Alexander Hamilton wrote that lotteries should be kept simple, and that "Everybody ... will be willing to hazard a trifling sum for the chance of considerable gain ... and would prefer a small chance of winning a great deal to a great chance of winning little." Taxes had never been accepted as a way to raise public funding for projects, and this led to the popular belief that lotteries were a form of hidden tax.

1776 Lottery ticket issued by Continental Congress to finance Revolutionary War.

At the end of the Revolutionary War the various states had to resort to lotteries to raise funds for numerous public projects. For many years these lotteries were highly successful and contributed to the nation's rapid growth. The lotteries were used for such diverse projects as the Pennsylvania Schuylkill  –  Susquehanna Canal (lottery in May 1795), and Harvard College (lottery in March 1806). Many American churches raised building funds through state authorized private lotteries. However, lotteries eventually became a cause of  financial mismanagement and scandal. Most notorious was the Louisiana State Lottery (1868 – 1892) 1892) which was aptly called the "Golden Octopus" because its tentacles reached into every home in America. Harvard Lottery Ticket 1811

Bolita, a type of lottery popular in Cuba, was brought to Tampa, Florida in the 1880s and flourished in Ybor City's many Latin saloons. Toward the end of the 19th century a large majority of  state constitutions banned lotteries. Finally, on July 29, 1890, U.S. President Benjamin Harrison sent a message Louisiana Lottery 1/20th of a $20 ticket: The Last of the Lotteries to Congress demanding "severe and effective legislation" against lotteries. Congress acted swiftly, and banned the U.S. mails from carrying lottery tickets. The Supreme Court upheld the law in 1892, and that brought a complete halt to all lotteries in the United States by 1900.

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When lotteries raised their head again in 1964, it would take many years of constitutional amendements by the various states before the lotteries were allowed to flourish again. On March 12, 1964, New Hampshire became the first state to sell lottery tickets in the modern era. For modern USA lotteries visit: Lotteries in the United States

New Hampshire Lottery Ticket 1964

Countries with a national lottery Africa • South South Africa Africa:: South South African African Nati Nationa onall Lottery Lottery • Kenya: Kenya: Toto Toto 6/49, 6/49, Kenya Kenya Charit Charityy Sweepst Sweepstake akess

North and South America • • • • • •

Argent Argentina ina:: Quiniel Quiniela, a, Loto Loto and variou variouss others others Barbad Barbados: os: Barba Barbados dos lotter lotteryy and vario various us others others Brazil Brazil:: MegaMega-Sen Senaa and and variou variouss others others Canada Canada:: Lotto Lotto 6/49 6/49 and and Lott Lottoo Max Max Chile: Chile: Polla Polla Chil Chilena ena de de Benefi Beneficen cencia cia S.A. S.A. Costa Costa Rica: Rica: Loterí Loteríaa Naciona Nacional, l, Chance Chancess Lotería Lotería Popular, Lotería Tiempos, and Lotería Instantanea (better known as "Raspaditas" since the tickets are scratch cards).

• El Salvador Salvador:: Lotería Lotería Nacional Nacional de Benefic Beneficencia encia,, Lotín (scratch cards). • Domini Dominican can Repub Republic lic:: Lotería Lotería Electr Electróni ónica ca Internacional Dominicana S.A. • Ecua Ecuado dor: r: Lot Loter ería ía Nac Nacio iona nall • Mexico: Mexico: Lotería Lotería Nacional Nacional para la Asistenc Asistencia ia Pública Pública and Pronósticos para la Asistencia Pública

This maneki neko beckons customers to purchase takarakuji tickets in Tokyo, Japan.

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Asia • • • • • • • • • • • • •

Mainland Mainland China: China: China Welfare Welfare Lottery, Lottery, China Sports Lottery Lottery Hong Hong Kong Kong:: Mar Markk Six Six Maca Macau: u: Maca Macauu SLO SLOT T Taiw Taiwan an:: Tai Taiwa wann Lot Lotte tery ry Isr Israel: Lo Lotto Japa Japan: n: Taka Takara raku kuji ji Leba Lebano non: n: La Lib Liban anai aise se des des Jeux Jeux Malaysia: Malaysia: Sports Sports Toto Malaysi Malaysia, a, Magnum Magnum Corporatio Corporation, n, Pan Malaysia Malaysiann Pools) Philip Philippin pines: es: Philip Philippin pinee Lotto Lotto Draw Draw Singa Singapo pore re:: Singa Singapo pore re Pool Poolss Sout Southh Kor Korea ea:: Lot Lotto to Sri Lanka: Lanka: National National Lottery, Lottery, Develo Development pment Lottery Lottery Thailand: สลากกินแบ่ นแบ่งรั งรั ฐบาล (salak gin bang ratthabarn or "Government Lottery"), also called lottery or (huay). • Viet Vietna nam: m: Xổ số kiến kiến thiế thiếtt

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Australasia • New New Zeal Zealan and: d: NZ Lot Lotte terie riess • Austra Australia lia:: OZ Lott Lotto, o, Power Powerbal ball, l, Lotto Lotto

Europe • • • • • • • • • • • •

PanPan-Eu Europ ropea ean: n: EuroMi EuroMill llio ions ns Nordic Nordic countr countries ies:: Viki Viking ng Lotto Lotto Austria: Austria: Lotto Lotto 6 aus 45, 45, EuroMill EuroMillions ions and Zahlenlotto Zahlenlotto Belgium: Belgium: Loterie Loterie National Nationalee or National Nationalee Loterij Loterij and EuroMillion EuroMillionss Bulgaria: Bulgaria: Durzha Durzhavna vna lotariy lotariya, a, TOTO TOTO 2 (6/49, (6/49, 6/42, 6/42, 5/35) 5/35) Croa Croati tia: a: Hrv Hrvat atsk skaa lutr lutrij ijaa Czec Czechh Rep Repub ublic lic:: Saz Sazka ka Denmar Denmark: k: Lotto, Lotto, Klasse Klasselot lotter teriet iet Finland: Finland: Lotto, Lotto, scratch scratch tickets, tickets, racing racing & football football pools pools (Veikkaus (Veikkaus)) France France:: La França Française ise des des Jeux and and EuroMi EuroMillio llions ns German Germany: y: Lotto Lotto 6 aus aus 49, 49, Spiel Spiel 77 and and Super Super 6 Gree Greece ce:: OPAP OPAP (Gr (Gree eek: k: ΟΠΑΠ – Οργανισμός Προγνωστικών  Αγώνων Ποδοσφαίρου ), Lotto 6/49, Joker 5/45 + 1/20 and various others

• Hung Hungaary: ry: Lot Lottó tó • Icel Icelan and: d: Lott Lottóó • Irelan Ireland: d: The The Natio National nal Lott Lottery ery (Iri (Irish: sh: An Chrannchur Náisiúnta ) and EuroMillions • Italy: Italy: Lotto Lotto and SuperE SuperEnal nalott ottoo • Latvia Latvia:: Latlot Latlotoo 5/35, 5/35, Super SuperBin Bingo, go, Keno Keno • Liechtenstei Liechtenstein: n: Internatio International nal Lottery Lottery in Liechtenste Liechtenstein in Foundation Foundation • Luxem Luxembo bour urg: g: Euro EuroMi Milli llion onss • Malta Malta:: Sup Super er 5 and and Lott Lottoo • Macedo Macedonia nia:: Lotar Lotarija ija na Make Makedon donija ija

A modern Finnish Lotto coupon, with personal info (customer no. and account for winnings) blanked out. These coupons are printed out on a terminal connected to the lottery provider (a monopoly, Veikkaus) whenever a player participates in the lottery.}

Lottery • • • • • • • • • • • • • • • • •

12 Monten Montenegr egro: o: Lutrij Lutrijaa Crne Crne Gore Gore Netherlands Netherlands:: Nationa Nationale le Postcod Postcodee Loterij, Loterij, Staatsloter Staatsloterij ij (The State Lottery) Norw Norway ay:: Lott Lottoo Pola Poland nd:: Lott Lottoo Portugal: Portugal: Lotaria Lotaria Clássica, Clássica, EuroMill EuroMillions ions and and Lotaria Lotaria Popular Popular Romani Romania: a: Loteri Loteriaa Română Română (6/4 (6/49, 9, 5/40, 5/40, Joker) Joker) Russia Russia:: Goslo Gosloto to (Rus (Russian sian:: Госло Гослото, то, The State Lottery) Serb Serb Republ Republic: ic: Lutr Lutrija ija Repu Republi blike ke Srpsk Srpskee Serbia Serbia:: Drža Državna vna Lutrij Lutrijaa Srbi Srbije je (The State Lottery of Serbia) Slo Slovaki vakia: a: Loto Loto Sloven Slovenia: ia: Loteri Loterija ja Sloven Slovenije ije Spain: Loterías Loterías y Apuesta Apuestass del Estado, Estado, EuroMillion EuroMillionss and ONCE • Catalonia: Catalonia: Loteria Loteria de Catalunya Catalunya (6/49 amongst amongst others) others) Swed Sweden en:: Lotto Lotto (Sve (Svensk nskaa Spel) Spel) Switze Switzerla rland: nd: Swiss Swiss Lott Lottoo and EuroM EuroMill illion ionss Turkey: Turkey: Various games games by Milli Piyango Piyango İdaresi (Nationa (Nationall Lottery Administra Administration) tion) including including Loto 6/49 6/49 and  jackpots Ukra Ukrain ine: e: Sup Super er Lot Lotto to United Kingdom Kingdom:: The National National Lottery Lottery,, the main game being being Lotto. Lotto. Also Monday Monday – The Charities Lottery and EuroMillions

Country lottery details In several countries, lotteries are legalized by the governments themselves. Several on-line lotteries and traditional lotteries with online payments exist. In the on-line lotteries, the user has to select their number and must either watch an ad for a few seconds before the selection is confirmed, or click on a web banner/link to register the pick in the system. The numbers may be drawn by the site that runs the online lottery or might be linked to a major physical lottery draw to ensure reliability. Prize money ranges from $100,000 to $100 million. [4]

Australia In Australia, lotteries operators are licensed at a state or territory level, and include both state government-owned and private sector companies.

Canada In Canada prior to 1967 buying a ticket on the Irish Sweepstakes was illegal. In that year the federal Liberal government introduced a special law (an Omnibus Bill) intended to bring up-to-date a number of obsolete laws. Pierre Trudeau, the Minister of Justice at that time, sponsored the bill. On September 12, 1967, Mr. Trudeau announced that his government would insert an amendment concerning lotteries. Even while the Omnibus Bill was still being written, Montreal mayor Jean Drapeau, trying to recover some of the money spent on the World ’s Fair and the new subway system, announced a "voluntary tax". For a $2.00 "donation" a player would be eligible to participate in a draw with a grand prize of $100 000. According to Drapeau, this "tax" was not a lottery for two reasons. The prizes were given out in the form of silver bars, not money, and the "competitors" chosen in a drawing would have to reply correctly to four questions about Montreal during a second draw. That competition would determine the value of the prize that the winner would win. The replies to the questions were printed on the back of the ticket and therefore the questions would not cause any undue problems. The inaugural draw was held on May 27, 1968.

Lottery There were debates in Ottawa and Quebec City about the legality of this 'voluntary tax'. The Minister of Justice alleged it was a lottery. Montreal ’s mayor replied that it did not contravene the federal law. While everyone awaited the verdict, the monthly draws went off without a hitch. Players from all over Canada, the United States, Europe, and Asia participated. On September 14, 1968 the Quebec Appeal Court declared Mayor Drapeau ’s "voluntary tax" illegal. However, the municipal authorities did not give up the struggle; the Council announced in November that the City would appeal this decision to the Supreme Court. As the debate over legalities continued, sales dropped significantly, because many people did not want to participate in anything illegal. Despite offers of new prizes the revenue continued to drop monthly, and by the nineteenth and final draw, was only a little over $800 000. On December 23, 1969 an amendment was made to the Canada's Criminal Code, allowing a provincial government to legally operate lottery systems. The first provincial lottery in Canada was Quebec's Inter-Loto in 1970. Other provinces and regions introduced their own lotteries through the 1970s, and the federal government ran Loto Canada (originally the Olympic Lottery) for several years starting in the late 1970s to help recoup the expenses of the 1976 Summer Olympics. Lottery wins are generally not subject to Canadian tax, but may be taxable in other jurisdictions, depending on the residency of the winner.[5] Today, Canada has two nation-wide lotteries: Lotto 6/49 and Lotto Max (the latter replaced Lotto Super7 in September of 2009). These games are administered by the Interprovincial Lottery Corporation, which is a consortium of the five regional lottery commissions, all of which are owned by their respective provincial and territorial governments: • • • •

Atlantic Atlantic Lottery Corporatio Corporationn (New Brunswick, Brunswick, Nova Scotia, Scotia, Prince Edward Island, Island, Newfoundland Newfoundland and Labrador) Labrador) LotoLoto-Qu Québ ébec ec (Que (Quebe bec) c) Ontario Ontario Lottery Lottery and Gaming Gaming Corpora Corporation tion (Ontario) (Ontario) Western Western Canada Lottery Lottery Corporation Corporation (Manitoba, (Manitoba, Saskatchew Saskatchewan, an, Alberta, Alberta, Yukon Territory, Territory, Northwest Northwest Territories, Territories, Nunavut) • British Columbia Columbia Lottery Corporation Corporation (British (British Columbia Columbia)) Primary, 48% of the total sales are used for jackpot, with the remaining 52% used for administration and sponsorship of hospitals and other local causes.

France The first known lottery in France was created by King Francis I in or around 1505. After that first attempt, lotteries were forbidden for two centuries. They reappeared at the end of the 17th century, as a "public lottery" for the Paris municipality (called   Loterie de L'Hotel de Ville) and as "private" ones for religious orders, mostly for nuns in convents. Lotteries quickly became became one of the most mos t important resources for religious congregations in the 18th century, and helped to build or rebuild about 15 churches in Paris, including St. Sulpice and Le Panthéon. At the beginning begi nning of the century, the King avoided having to fund religious orders by giving them the right to run lotteries, but the amounts generated became so large that the second part of the century turned into a struggle between the monarchy and the Church for control of the lotteries. In 1774, the monarchy —specifically Madame de Pompadour--founded the  Loterie de L'École Militaire to buy what is called today the Champ de Mars in Paris, and build a military academy that Napoleon Bonaparte would later attend; they also banned all other lotteries, with 3 or 4 minor exceptions. This lottery became known a few years later as the  Loterie Royale de France. Just before the French Revolution in 1789, the revenues from La Lotterie Royale de France were equivalent to between 5 and 7% of total French revenues.

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Lottery Throughout the 18th century, philosophers like Voltaire as well as some bishops complained that lotteries exploit the poor. This subject has generated much oral and written debate over the morality of the lottery. All lotteries (including state lotteries) were frowned upon by idealists of the French Revolution, who viewed them as a method used by the rich for cheating the poor out of their wages. The Lottery reappeared again in 1936, called lotto, when socialists needed to increase state revenue. Since that time,   La Française des Jeux (government owned) has had a monopoly on most of the games in France, including the lotteries. There have also been reports of lotteries regarding the mass guillotine executions in France. It has been said that a number was attached to the head of each person to be executed and then after all the executions, the executioner would pull out one head and the people with the number that matched the one on the head were awarded prizes (usually small ones); each number was 3-to-5 digits long.

Liechtenstein The International Lottery in Liechtenstein Foundation (ILLF) is a government authorised and state controlled charitable foundation that operates Internet lotteries. The ILLF pioneered Internet gaming, having launched the web’s first online lottery, PLUS Lotto, in 1995 and processed the first online gaming transaction ever. The International Lottery in Liechtenstein Foundation (ILLF) also introduced the first instant scratchcard games on the Internet during this time. The ILLF supports the International Federation of Red Cross and Red Crescent Societies and other charitable causes in Liechtenstein, many of which support projects in poorer nations internationally. The ILLF operates many websites, referred to as the ILLF brands. Combined, these brands offer a wide array of  games to choose from. Lottery winnings are not taxed in Liechtenstein.

New Zealand Lotteries in New Zealand are controlled by the Government. A state owned trading organisation, the New Zealand Lotteries Commission, operates low prize scratch ticket games and Powerball type lotteries with weekly prize   jackpots. Lottery profits are distributed by the New Zealand Lottery Grants Board directly to charities and community organisations. Sport and Recreation New Zealand, Creative New Zealand and the New Zealand Film Commission are statutory bodies that operate autonomously in distributing their allocations from the Lottery Grants Board. The lotteries are drawn on Saturday and Wednesday. Lotto is sold via a network of computer terminals in shopping centers across the nation. The Lotto game was first played in 1987 and replaced New Zealand's original national lotteries, the Art Union and Golden Kiwi. Lotto is a pick 6 from 40 numbers game. The odds of winning the first division prize of around NZ$300,000 to NZ$2 million are 1 in 3,838,380. The Powerball game is the standard pick 6 from 40 Lotto numbers with an additional pick 1 from 10 Powerball number. This game has odds of 1 in 38,383,800 and a first prize of between NZ$1 million and NZ$30 million [6] . In 2007 Powerball changed to a pick 1 of 10 game (formerly pick 1 of 8) and the minimum Powerball prize increased from $1 million to $2 million. Big Wednesday is a game played by picking 6 numbers from 45 plus heads or tails from a coin toss. A jackpot cash prize of NZ$1 million to NZ$15 million is supplemented with product prizes such as Porsche and Aston Martin cars, boats, holiday homes and luxury travel. The odds of winning first prize are 1 in 16,290,120. Website operators independent of the state Lotteries Commission[7] began publishing online Lotto results[8] as early as 1998.[9] An interactive Lotto website authorised to sell tickets online was established in 2007. Lottery winnings are not taxed in New Zealand.

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Lottery

United States In the United States, the existence of lotteries is subject to the laws of each state; there is no national lottery. Private lotteries were legal in the United States in the early 1800s. [10] In fact, a number of US patents were granted on new types of lotteries. In today's vernacular, these would be considered business method patents. Before the advent of state-sponsored lotteries, many illegal lotteries thrived; for example, see Numbers game and Peter H. Header from 1840 US patent on a new type of private lottery Matthews. The first modern state lottery in the U.S. was established in the state of New Hampshire in 1964; as of 2008, lotteries are established in 42 states, the District of Columbia, Puerto Rico, and the Virgin Islands; Arkansas voters, on November 4, 2008, approved a ballot question to legalize a state lottery. The first modern interstate lottery in the U.S. was formed in 1985 and linked Maine, New Ha mpshire and and Vermont. Vermont. In 1988, the Multi-State Lottery Association (MUSL) was formed with Oregon, Iowa, Kansas, Rhode Island, West Virginia, Missouri, and the District of Columbia as its charter members; it is best known f or or its "Powerball" drawing, drawing, which is designed to build up very large jackpots.As of May 2010 the following states are part of powerball. Arizona, Arkansas, Colorado, Connecticut, Delaware, D.C. Florida, Georgia, Idahdo, Illinois, Indiana, Iowa, Kansas, Kentucky, Louisiana, Maine, Maryland, Massachusetts, Michigan, Miniestoa,Missouri, Montana, Nebraska, New Hampshire, New Jersey, New Mexico, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania, Rhode Islan d, South Carolina, South Dakota, Tennessee, Texas, Vermont, Virgin Islands, Virgina, Washington, West Virgina and Wisconsin. Another interstate lottery, The Big Game (now called Mega Millions), was formed in 1996 by the states of Georgia, Illinois, Massachusetts, Maryland, Michigan and Virginia as its charter members. These states were joined by New Jersey (1999), New York and Ohio (May 2002), Washington state (September 2002), Texas (2003) and California (2005) for a total of 12 members. [11] Instant lottery tickets, also known as scratch cards, were first introduced in the 1970s and have since become a major source of state lottery revenue. Some states have introduced keno and video lottery terminals (slot machines in all but name). Other interstate lotteries include Cashola, Hot Lotto and Wild Card 2, some of MUSL's other games. With the advent of the Internet it became possible for people to play lottery-style games on-line, many times for free (the cost of the ticket being supplemented by merely seeing an ad or some other form of revenue). GTech Corporation, in the United States, administers 70% of the worldwide online and instant lottery business, according to its website. With online gaming rules generally prohibitive, "lottery" games face less scrutiny. This is leading to the increase in web sites offering lottery ticket purchasing services, charging premiums on base lottery prices. The legality of such services falls into question across many jurisdictions, especially throughout the United States, as the gambling laws related to lottery play generally have not kept pace with the spread of technology. An evolution of the lottery on the internet has appeared on the social network Facebook. The free lottery has weekly drawings and allows people to receive daily lottery tickets and send their friends tickets. Presently, large portions of many American state lotteries are used to fund public education systems.

15

Lottery

Probability of winning The chances of winning a lottery jackpot are determined by several factors, including: the count of possible numbers, the count of winning numbers drawn, whether or not order is significant and whether drawn numbers are returned for the possibility of further drawing. In a typical 6 from 49 lotto, 6 numbers are drawn from 49 and if the 6 numbers on a ticket match the numbers drawn, the ticket holder is a jackpot winner  – this is true regardless of the order in which the nu mbers are drawn. drawn. The odds of being a jackpot winner are approximately 1 in 14 million (13,983,816 to be exact). The derivation of this result (and other winning scores) is shown in the Lottery mathematics article. To put these odds in context, suppose one buys one lottery ticket per week. 13,983,816 weeks is roughly 269,000 years; In the quarter-million years of play, one would expect to win the jackpot only once, or if one person bought a ticket every second of every day for one year, one would win the jackpot on average about 2.25 times. The odds of winning any actual lottery can vary widely depending on the lottery design of financial engineers. Mega Millions is a very popular multi-state lottery in the United States which is known for jackpots that grow very large from time to time. This attractive feature is made possible simply by designing the game to be extremely difficult to win: 1 chance in 175,711,536. That's over twelve times higher than the example above. Mega Millions players also pick six numbers, but two different diff erent "bags" are used. The first five numbers come from one bag that contains numbers from 1 to 56. The sixth number  –  the "Mega Ball number"  –  comes from the second bag, bag, which contains contains numbers from 1 to 46. To win a Mega Millions jackpot, a player's five regular numbers must match the five regular numbers drawn and the Mega Ball number must match the Mega Ball number drawn. In other words, it is not good enough to pick 10, 18, 25, 33, 42 / 7 when the drawing is 7, 10, 25, 33, 42 / 18. Even though the player picked all the right numbers, the Mega Ball number at the end of the ticket doesn't match the one drawn, so the ticket would be credited with matching only four numbers (10, 25, 33, 42). The SuperEnalotto of Italy is supposedly the most difficult, as players try to match 6 numbers out of 90. The odds in making the jackpot: 1 in 622,614,630. Most lotteries give lesser prizes for matching just some of the winning numbers. The Mega Millions game is an extreme case, giving a very small payout (US$2) even if a player matches only the final Mega Ball number number on the ticket. The weekly 6/49 lottery operated by the ILLF, offers a two ball cash prize to make the odds of winning any prize only 1 in 6.63. Matching more numbers, the payout goes up. Although none of these additional prizes affect the chances of winning the jackpot, they do improve the odds of winning something and therefore add a little to the value of the ticket. In most lotteries, if a large amount of smaller prizes are awarded, the jackpot will be reduced, in a similar manner that the jackpot is divided if multiple players have tickets with all the winning numbers. In the UK National Lottery the smallest prize is £10 for matching three balls. There exists a Wheeling Challenge [12] to create the smallest set of tickets to cover enough combinations to ensure that any 6 numbers drawn will match against at least 3 numbers on at least one of the tickets. The current record is 163 tickets. The expected value of lottery bets is often notably low. In the United States, an expected value of 50% of the purchase price is common. For instance, when the player buys a lottery ticket for, say, $10 he obtains a financial asset with an expected value of only $5. Hence, buying a lottery ticket reduces the buyer's expected net worth. This is in contrast with financial securities like stocks and bonds whose prices are theoretically based on their expected real values, as expected by the markets at any given point in time. Lotteries are sometimes described as a regressive tax, albeit a voluntary one, since those most likely to buy tickets, and to spend a larger proportion of their money on them, are typically less affluent people. The astronomically high odds against winning the larger prizes have also led to the epithets of a "tax on stupidity" and a "math tax". Although the use of the word "tax" is not strictly correct, these descriptions are intended to suggest that lotteries are government-sanctioned operations which will attract only those people who fail to understand that buying a lottery ticket is a poor economic decision. Indeed, after taking into account the present value of a given lottery prize as a

16

Lottery

17

single lump sum cash payment, the impact of any taxes that might apply, and the likelihood of having to share the prize with other winners, it is not uncommon to find that a ticket for a major lottery is worth less than one third of its purchase price. In other words, if a lottery ticket costs US$1 to purchase, its true economic worth may be only US$0.33 or so at the time of purchase. Of course, this is just a hypothetical example, and the actual value will depend on the details of each lottery. Some lotteries may offer tickets that are worth less than 20% of their price, while others may be worth over 50%. To raise money, lottery operators must offer tickets worth much less than what one pays for them, so the lottery is a bad choice for customers trying to come out ahead. In a famous occurrence, a Polish-Irish businessman named Stefan Klincewicz bought up almost all of the 1,947,792 combinations available on the Irish lottery. He and his associates paid less than one million Irish pounds while the   jackpot stood at £1.7 million. There were three winning tickets, but with the "Match 4" and "Match 5" prizes, Klincewicz made a small profit overall.

Notable prizes Prize (local currency)

Lottery

Country

Winner

Date

Notes

$390m

Mega Millions

United States

Won by one ticket holder from New Jersey and one from Georgia

6 March 2007

$365m

Powerball

United States

One ticket bought jointly by eight co-workers at a Nebraska meat processing plant

18 February World's largest 2006 single ticket winner

$363m

The Big Game

United States

Two winning tickets: Larry and 9 May 2000 The Big Game is Nancy Ross (Michigan), Joe now named Mega and Sue Kainz (Illinois) Millions

United United States

Andrew Jackson “Jack” Whittaker, Jr.

2002-12-25

World's largest single person winner

$314.9 $314.9 Millio Millionn Powerb Powerball all

World's largest  jackpot

€180m

EuroMillions

France ×2, Portugal ×1

Three ticket holders

3 February 2006

Europe's largest  jackpot

€147,8m

SuperEnalotto

Italy

One ticket holder from Bagnone 22 August (Toscana) 2009

Europe's largest single winner

€126m

EuroMillions

Spain

Anonymous 25 year-old woman 8 May 2009 La Largest single winner in EuroMillions.

£42m

National Lottery

€37.6m

National Lottery

Germany

€25m

State Lottery

€115m

Euro Millions

€42m

United Kingdom

Three ticket holders

6 January 1996

Won by a nurse from North Rhine-Westphalia

7 October 2006

Largest German prize and single winner

Netherlands Ticket sold in The Hague

10 July 2008

Tax free lump sum

Ireland

Dolores McNamara

August 2005 Biggest single winner and jackpot (Ireland) Tax free lump sum

Jordan Banks

August 2008 Tax free lump sum

Lottery ₱347.8m

18 Phili Philipp ppin ines es

Two Two winn winner erss in Luzo Luzonn

(US$4.765m)

Philippine Lotto Draw

22 Febr Februa uary ry Asia's largest prize 2009

R$145m

Mega-Sena

Brazil

Won by one ticket holder from Brasília (Federal District) and one from Santa Rita do Passa Quatro (São Paulo)

31 December 2009

South America's largest prize

£56m

EuroMillions

Nigel Page and his partner Justine Laycock. The total  jackpot of £113m was shared with a winner in Spain.

12 January 2010

Britain's biggest ever lottery prize

United Kingdom

Sources:

http://www.usamega. http://www. usamega.com/archive-052000. com/archive-052000.htm htm http://www.timesonline. http://www. timesonline.co. co.uk/tol/news/world/europe/article6274441. uk/tol/news/world/europe/article6274441. ece http://news.bbc. http://news. bbc.co. co.uk/1/hi/world/europe/4746057.stm uk/1/hi/world/europe/4746057.stm http://news.bbc. http://news. bbc.co. co.uk/1/hi/uk/4676172. uk/1/hi/uk/4676172. stm http://news.bbc. http://news. bbc.co. co.uk/1/hi/world/americas/4740982. uk/1/hi/world/americas/4740982. stm http://www.sisal. http://www. sisal.it/se/se_main/1,4136,se_Record_Default,00. it/se/se_main/1,4136,se_Record_Default,00.html html http://www.howtomakeabilliondollars. http://www. howtomakeabilliondollars.com/145-million-european-lottery-this-weekend/  com/145-million-european-lottery-this-weekend/  http://www.gelderlander. http://www. gelderlander. nl/algemeen/dgbinnenland/3405786/Jackpot-van-25-miljoen-valt-in-regio-Den-Haag. ece On 20 September 2005 a primary school boy in Italy won the equivalent of £27.6 million in the Italian national lottery. Although children are not allowed to gamble under Italian law, children are allowed to play the lottery. [13]

Payment of prizes Winnings (in the U.S.) are not necessarily paid out in a lump sum, contrary to the expectation of many lottery participants. In certain countries, mainly the U.S., the winner gets to choose between an annuity payment and a one-time payment. The one-time payment is much smaller than, indeed often only half of, the advertised lottery  jackpot, even before applying any withholdings to which the prize may be subject. While taxes vary by state and how on winnings are invested, a rough rule of thumb is that a winner who takes a lump sum can reasonably expect to pocket 1/3 of the jackpot amount after the initial tax withholding and additional taxes at the end of the tax year. Therefore, a winner of a $100,000,000 jackpot who takes a lump sum can roughly expect to have $33,000,000 after filing income tax documents for the year in which the jackpot was won. The annuity option provides regular payments over a period that ranges from 10 to 40 years. Some U.S. lottery games, especially those offering a "lifetime" prize, do not offer a lump-sum option. In some online lotteries, the annual payments can be as little as $25,000 over 40 years, with a balloon payment in the final year. This type of installment payment is often made through investment in government-backed securities. Online lotteries pay the winners through their insurance backup. However, many winners choose to take the lump-sum payment, since they believe they can get a better rate of return on their investment elsewhere. In some countries, lottery winnings are not subject to personal income tax, so there are no tax consequences to consider in choosing a payment option. In Canada, Australia, Germany, Ireland, Italy and the United Kingdom all prizes are immediately paid out as one lump sum, tax-free to the winner. In Liechtenstein, all winnings are tax-free and the winner may opt to receive a lump sum or an annuity with regard to the Jackpot prizes. In the United States, federal courts have consistently held that lump sum payments received from third parties in exchange for the rights to lottery annuities are not capital assets for tax purpose. Rather, the lump sum is subject to ordinary income tax treatment.

Lottery

19

Problems Side-effects There can be some problems associated with winning a lottery jackpot. Those of a poor socioeconomic background may not have proper money management skills. In addition, there are security and safety risks associated with publicly announcing the lottery winners such as holding family members for ransom. Winners some times feel anomie from the dramatic change of lifestyles.

Scams and frauds Lottery, like any form of gambling, is susceptible to fraud, despite the high degree of scrutiny claimed by the organizers. One method involved is to tamper with the machine used for the number selection. By rigging a machine, it is theoretically easy to win a lottery. This act is often done in connivance with an employee of the lottery firm. Methods used vary; loaded balls where select balls are made to pop-up making it either lighter or heavier than the rest. All balls should be independently verified for materials, size, pressure, susceptibility to magnetism, and other qualities. In some US States, such as Kansas and Minnesota, losing lottery tickets can be mailed in for a raffle of special prizes. The trouble with that is that employees of stores that sell lottery tickets sometimes collect the lottery tickets that are thrown away and send them in. As a lottery official put it "The retailers have an unlimited supply of free tickets. You do not need to be an FBI agent to realize that is a tremendously unfair advantage." [14] Some advance fee fraud scams on the Internet are based on lotteries. The fraud starts with spam congratulating the recipient on their recent lottery win. The email explains that in order to release funds the email recipient must part with a certain amount (as tax/fees) as per the rules or risk forfeiture. Another form of lottery scam involves the selling of "systems" which purport to improve a player's chances of  selecting the winning numbers in a Lotto game. These scams are generally based on the buyer's (and perhaps the seller's) misunderstanding of probability and random numbers. Sale of these systems or software is legal, however, since they mention that the product cannot guarantee a win, let alone a jackpot. Another famous scam was the ORS World Cup Sweepgate scandal. In which a supposedly random draw ended with the competition organiser Michael Davies, having almost 25% chance of winning where others had as little as 0.5% chance. Although Mr Davies has denied any such fix, the facts seem clear and he looks certain to be brought to  justice when the truth is finally revealed.

See also • • • • • • •

Betting po pool Comb Combin inat ator oria iall numbe numberr system system Gami Gaming ng mat mathe hema mati tics cs GTec GTechh Cor Corpo pora rati tion on Intralot Keno Lott Lotter eryy Whee Wheeli ling ng

Lottery

20

Further reading • • • • •

A History History of English English Lotteries, Lotteries, by John John Ashton, Ashton, London: London: Leadenhal Leadenhalll Press, 1893 1893 Fortune's Fortune's Merry Merry Wheel, Wheel, by John Samuel Samuel Ezell, Ezell, Harvard Harvard Universi University ty Press, 1960. Lotteries Lotteries and and Sweeps Sweepstakes, takes, 1932 by Ewen L'Estrange L'Estrange The Lottery Lottery Encyclo Encyclopedia, pedia, 1986 by Ron Shelley Shelley (NY Public Public Library) Library) Fate's Fate's Bookie: Bookie: How The Lottery Lottery Shaped Shaped The The World World by Gary Hicks, Hicks, History History Press, Press, 2009 2009

External links • Worl Worldd Lott Lotter eryy Asso Associ ciat atio ionn [15] • Euler's Euler's Anal Analysi ysiss of the the Genoe Genoese se Lotte Lottery ry [16]

References [1] [2] [3] [4]

Ron Shelley, Shelley, The The Lottery Lottery Encyclopedi Encyclopedia(1986) a(1986) John Ashton, Ashton, A History History of English English Lotteries Lotteries,, 1893. John Samuel Samuel Ezell, Ezell, Fortune's Fortune's Merry Merry Wheel, 1960. 1960. Two Winners to Share Record Record A$106 Million Australian Australian Lottery (http://www.bloo (http://www.bloomberg. mberg.com/apps/news?pid=20601081& com/apps/news?pid=20601081& sid=ahKl7APqrtD0) [5] Internet Internet Archive Archive (http://web. (http://web.archive. archive.org/web/20060423062311/http://lotteries. org/web/20060423062311/http://lotteries. olgc.ca/consumer_fq. olgc.ca/consumer_fq. jsp#qa17)  jsp#qa17) [6] Prize Divisions Divisions (http://ww (http://www.mylott w.mylotto. o.co. co.nz/wps/wcm/myconnect/lotteries2/nzlotteries/Primary/Our_Games/Lotto/AllAboutLotto/  nz/wps/wcm/myconnect/lotteries2/nzlotteries/Primary/Our_Games/Lotto/AllAboutLotto/  LottoPrizeDivisions. jsp)  jsp) [7] New Zealand Zealand State Lotterie Lotteriess Commission Commission (http://www. (http://www.mylotto. mylotto.co. co.nz/wps/wcm/myconnect/lotteries2/nzlotteries/Global/  nz/wps/wcm/myconnect/lotteries2/nzlotteries/Global/  AboutNZLotteries/StatutoryFunction/) [8] New Zealand Zealand Lotto Lotto Results Results (http://lot (http://lotto. to.nzpages. nzpages.net. net.nz) nz) [9] Internet Internet Archive Archive (http://web. (http://web.archive. archive.org/web/19981212015337/http://lotto.nzpages. org/web/19981212015337/http://lotto.nzpages. net.nz/) net.nz/) [10] Bellho Bellhouse use,, D.R., D.R., “The Genoese Lottery”, Statistical Science, vol. 6, No. 2. (May, 1991), pp. 141 -148 [11] Megamillions Megamillions game history history (http://www. (http://www.megamillions. megamillions.com/aboutus/game_history. com/aboutus/game_history.asp) asp) [12] http://l http://lott ottery. ery.merseyworld. merseyworld.com/Wheel/Wheel.html com/Wheel/Wheel.html [13] [13] http http:// ://ww www. w.dailyrecord. dailyrecord.co. co.uk/news/tm_objectid=16164112&method=full&siteid=66633& uk/news/tm_objectid=16164112&method=full&siteid=66633& headline=primary-pupil-s--pound-27m-lotto-win--name_page.html headline=primary-pupil-s--pound-27m-lotto-win--name_page. html [14] "Legalized "Legalized Gambling; Gambling; America's America's Bad Bad Bet" by John Eidsmoe Eidsmoe [15] [15] http http:// ://ww www. w.world-lotteries. world-lotteries.org/  org/  [16] http://m http://math athdl. dl.maa. maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=217&bodyId=93 org/convergence/1/?pa=content&sa=viewDocument&nodeId=217&bodyId=93

Combinatorial number system

21

Combinatorial Combinatorial number system In mathematics, and in particular in combinatorics, the combinatorial number system of degree k  (for some positive integer k ), ), also referred to as combinadics, is a correspondence between natural numbers (taken to include 0)  N  and k -combinations, -combinations, represented as strictly decreasing sequences ck  > ... > c2 > c1 ≥ 0. Since the latter are strings of numbers, one can view this as a kind of numeral system for representing  N , although the main utility is representing a k -combination -combination by  N  rather than the other way around. Distinct numbers correspond to distinct k -combinations, -combinations, and produce them in lexicographic order; moreover the numbers less than correspond to all

k -combinations -combinations of { 0, 1, ..., n − 1}. The correspondence does not depend on the size n of the set that the k -combinations -combinations are taken from, so it can be interpreted as a map from N to the k -combinations -combinations taken from N; in this view the correspondence is a bijection. The number N corresponding to (ck ,...,c2,c1) is given by

The fact that a unique sequence so corresponds to any number  N  was observed by D.H. Lehmer.[1] Indeed a greedy algorithm finds the k -combination -combination corresponding to  N : take ck  maximal with , then take ck  − 1 maximal with

, and so forth. The originally used term "combinatorial representation of integers" is

shortened to "combinatorial number system" by Knuth,[2] who also gives a much older reference; [3] the term "combinadic" is introduced by James McCaffrey [4] (without reference to previous terminology or work). Unlike the factorial number system, the combinatorial number system of degree k  is not a mixed radix system: the part of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value. The main application of the combinatorial number system is that it allows rapid computation of the k -combination -combination that is at a given position in the lexicographic ordering, without having to explicitly list the k -combinations -combinations preceding it; this allows for instance random generation of  k -combinations -combinations of a given set. Enumeration of  k -combinations -combinations has many applications, among which software testing, sampling, quality control, and the analysis of  lottery games.

Ordering combinations A k -combination -combination of a set S is a subset of  S with k (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all possible k -combinations -combinations of a set S  of  n elements. Choosing, for any n, {0, 1, ..., n − 1} as such a set, it can be arranged that the representation of a given k -combination -combination C  is independent of the value of  n (although n must of course be sufficiently large); in other words considering C  as a subset of a larger set by increasing n will not change the number that represents C . Thus for the combinatorial number system one just considers C  as a k -combination -combination of the set N of all natural numbers, without explicitly mentioning n. In order to ensure that the numbers representing the k -combinations -combinations of {0, 1, ..., n − 1} are less than those representing k -combinations -combinations not contained in {0, 1, ..., n − 1}, the k -combinations -combinations must be ordered in such s uch a way that their largest elements are compared first. The most natural ordering that has this property is lexicographic ordering of the decreasing sequence of their elements. So comparing the 5-combinations C = {0,3,4,6,9} and C ' = {0,1,3,7,9}, one has that C  comes before C ',', since they have the same largest part 9, but the next largest part 6 of  C  is less than the next largest part 7 of  C ';'; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0). Another way to describe this ordering is view combinations as describing the k  raised bits in the binary representation of a number, so that C = {c1,...,ck } describes the number

Combinatorial number system

22

(this associates distinct numbers to all finite sets of natural numbers); then comparison of  k -combinations -combinations can be done by comparing the associated binary numbers. In the example C  and C ' correspond to numbers 10010110012 = 60110 and 10100010112 = 65110, which again shows that C  comes before C '.'. This number is not however the one one wants to represent the k -combination -combination with, since many binary numbers have a number of raised bits different form k ; one wants to find the relative position of  C in the ordered list of (only) k -combinations. -combinations.

Place of a combination in the ordering The number associated in the combinatorial number system of degree k  to a k -combination -combination C  is the number of  k -combinations -combinations strictly less than C  in the given ordering. This number can be computed from C = { ck , ..., c2, c1 } with ck  > ... > c2 > c1 as follows. From the definition of the ordering it follows that for each k -combination -combination S  strictly less than C , there is a unique index i such that ci is absent from S , while ck , ..., ci+1 are present in S , and no other value larger than ci is. One can therefore group those k -combinations -combinations S according to the possible values 1, 2, ..., k of  i, and count each group separately. For a given value of  i one must include ck , ..., ci+1 in S , and the remaining i elements of  S  must be chosen from the ci non-negative integers strictly less than ci; moreover any such choice will result in a k -combinations -combinations S  strictly less than C . The number of possible choices is , which is therefore the number of combinations in group i; the total number of  k -combinations -combinations strictly less than C then is

and this is the index (starting from 0) of  C  in the ordered list of  k -combinations. -combinations. Obviously there is for every  N ∈ N exactly one k -combination -combination at index N  in the list (supposing k ≥ 1, since the list is then infinite), so the above argument proves that every  N  can be written in exactly one way as a sum of  k  binomial coefficients of the given form.

Finding the  k-combination for a given number The given formula allows finding the place in the lexicographic ordering of a given k -combination -combination immediately. The reverse process of finding the k -combination -combination at a given place  N requires somewhat more work, but is straightforward nonetheless. By the definition of the lexicographic ordering, two k -combinations -combinations that differ in their largest element ck  will be ordered according to the comparison of those largest elements, from which it follows that all combinations with a fixed value of their largest element are contiguous in the list. Moreover the smallest combination with ck  as largest element is , and it has ci = i − 1 for all i < k  (for this combination all terms in the expression except are zero). Therefore ck  is the largest number such that . If  k > 1 the remaining elements of the k -combination -combination form the k  − 1-combination corresponding to the number in the combinatorial number system of degree k  − 1, and can therefore be found by continuing in the same way for

and k  − 1 instead

 N and k . of  N  Example Suppose one wants to determine the 5-combination at position 72. The successive values of

for n = 4, 5, 6, ...

are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for n = 8. Therefore c5 = 8, and the remaining elements form the 4-combination at position 72 − 56 = 16. The successive values of for n = 3, 4, 5, ... are 0, 1, 5, 15, 35, ..., of which the largest one not exceeding 16 is 15, for n = 6, so c4 = 6. Continuing similarly to search for a 3-combination at position 16 − 15 = 1 one finds c3 = 3, which uses up the final unit; this establishes , and the remaining values ci will be the maximal ones with , namely ci = i − 1. Thus we have found the 5-combination {8, 6, 3, 1, 0}.

Combinatorial number system

23

Applications One could use the combinatorial number system to list or traverse all k -combinations -combinations of a given finite set, but this is a very inefficient way to do that. Indeed, given some k -combination -combination it is much easier to find the next combination in lexicographic ordering directly than to convert a number to a k -combination -combination by the method indicated above. To find the next combination, find the smallest i ≥ 2 for which ci ≥ ci− 1+2; then increase ci−1 by one and set all c j with j < i − 1 to their minimal value  j − 1. If the k -combination -combination is represented as a binary value with k bits 1, then the next such value can be computed without any loop using bitwise arithmetic: the following function will advance x to that value or return false: // find next k-combination bool next_combination( next_com bination(u unsigned long long& & x) // assume x has form x'01^a10 ^b in binary { unsigned long u = x & -x; // extract rightmost bit 1; u =

0'00^a10^b

unsigned long v = u + x; // set last non-trailing bit 0, and clear to the right; v= v =x'10^ x'10^a00 a00^ ^b if

(v==0 (v==0) ) // then overflow in v, or x==0

return false ;

// signal that next k-combination cannot be

represented x = v +(((v^ (((v^x)/ x)/u)>>2 u)>>2); ); // v^x = 0'11^a10^b, (v^x)/u = 0'0^b1^{a+2}, and  x ← x'100^ x'100^b1 b1^ ^a return true;

// successful completion

}

This is called Gosper's hack; [5] corresponding assembly code was described as item 175 in HAKMEM. On the other hand the possibility to directly generate the k -combination -combination at index  N  has useful applications. Notably, it allows generating a random k -combination -combination of an n-element set using a random integer  N  with , simply by converting that number to the corresponding k -combination. -combination. If a computer program needs to maintain a table with information about every k -combination -combination of a given finite set, the computation of the index  N associated to a combination will allow the table to be accessed without searching.

See also • Factorial Factorial number number system system (also called called factorad factoradics) ics)

References [1]  Applied Combinatorial Mathematics , Ed. E. F. Beckenbach (1964), pp.27−30. [2] Knuth, D. E. (2005), "Generati "Generating ng All Combinations Combinations and Partitions" Partitions",, The Art of Computer Programming, 4, Fascicle 3, Addison-Wesley, pp. 5−6, ISBN 0-201-85394-9. [3] Pascal Pascal,, Ernesto Ernesto (1887), (1887), Giornale di Matematiche , 25, pp. 45−49 Combination (http://msdn.microsoft. [4] McCaff McCaffrey rey,, James (2004) (2004),, Generating the mth Lexicographical Element of a Mathematical Combination (http://msdn. microsoft.com/  com/  en-us/library/aa289166(VS.71). en-us/library/aa289166(VS. 71).aspx), aspx), Microsoft Developer Network, Programming , 4, Fascicle 1, Addison-Wesley, pp. 54, [5] Knuth, D. E. (2009), (2009), "Bitwise "Bitwise tricks tricks and techniques", techniques", The Art of Computer Programming ISBN 0-321-58050-8

Gaming mathematics

Gaming mathematics Gaming mathematics, also referred to as the mathematics of gambling, is a collection of probability applications

encountered in games of chance and can be included in applied mathematics. From mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events.

Experiments, events, probability spaces The technical processes of a game stand for experiments that generate aleatory events. Here are few examples: • Throwing Throwing the dice in craps craps is an experiment experiment that that generates generates events events such as occurrenc occurrences es of certain certain numbers numbers on the dice, obtaining a certain sum of the shown numbers, obtaining numbers with certain properties (less than a specific number, higher that a specific number, even, uneven, and so on). The sample space of such an experiment is {1, 2, 3, 4, 5, 6} for rolling one die or {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), (6, 2), ..., (6, 6)} for rolling two dice. The latter is a set of ordered pairs and counts 6 x 6 = 36 elements. The events can be identified with sets, namely parts of the sample space. For example, the event occurrence of an even number  is represented by the following set in the experiment of rolling one die: {2, 4, 6}. • Spinning Spinning the roulette roulette wheel is an experiment experiment whose whose generated generated events events could be the occurrence occurrence of a certain certain number, of a certain color or a certain property of the numbers (low, high, even, uneven, from a certain row or column, and so on). The sample space of the experiment involving spinning the roulette wheel is the set of numbers the roulette holds: {1, 2, 3, ..., 36, 0, 00} for the American roulette, or {1, 2, 3, ..., 36, 0} for the European. The event occurrence of a red number  is represented by the set {1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}. These are the numbers inscribed in red on the roulette wheel and table. • Deal Dealin ingg car cards ds in blackjack is an experiment that generates events such as the occurrence of a certain card or value as the first card dealt, obtaining a certain total of points from the first two cards dealt, exceeding 21 points from the first three cards dealt, and so on. In card games we encounter many types of experiments and categories of  events. Each type of experiment has its own sample space. For example, the experiment of dealing the first card to the first player has as its sample space the set of all 52 cards (or 104, if played with two decks). The experiment of  dealing the second card to the first player has as its sample space the set of all 52 cards (or 104), less the first card dealt. The experiment of dealing the first two cards to the first player has as its sample space a set of ordered pairs, namely all the 2-size arrangements of cards from the 52 (or 104). In a game with one player, the event the  player is dealt a card of 10 points as the first dealt card is represented by the set of cards {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦, K♠, K♣, K♥, K♦}. The event the player is dealt a total of five points from the first two dealt cards is represented by the set of 2-size combinations of card values {(A, 4), (2, 3)}, which in fact counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and symbol). • In 6/49 lottery, lottery, the experime experiment nt of drawing drawing six numbers numbers from the 49 generate generate events events such as drawing drawing six specific specific numbers, drawing five numbers from six specific numbers, drawing four numbers from six specific numbers, drawing at least one number from a certain group of numbers, etc. The sample space here is the set of all 6-size combinations of numbers from the 49. • In draw poker, poker, the experiment experiment of dealing dealing the initial five five card hands generate generatess events such as dealing dealing at least one certain card to a specific player, dealing a pair to at least two players, dealing four identical symbols to at least one player, and so on. The sample space in this case is the set of all 5-card combinations from the 52 (or the deck used). • Dealing Dealing two cards to a player player who has has discarded discarded two cards cards is another experime experiment nt whose sample sample space space is now the set of all 2-card combinations from the 52, less the cards seen by the observer who solves the probability problem. For example, if you are in play in the above situation and want to figure out some odds regarding your hand, the

24

Gaming mathematics sample space you should consider is the set of all 2-card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2-size combinations from 47.

The probability model A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the space (field) of events. The event is the main unit probability theory works on. In gambling, there are many categories of events, all of which can be textually predefined. In the previous examples of gambling experiments we saw some of the events that experiments generate. They are a minute part of all possible events, which in fact is the set of all parts of the sample space. For a specific game, the various types of events can be: • Events Events relat related ed to your your own own play play or to to oppone opponents nts’ play; • Event Eventss rela relate tedd to one one per perso sonn’s play or to several persons ’ play; • Immedia Immediate te even events ts or long long-sh -shot ot event events. s. Each category can be further divided into several other subcategories, depending on the game referred to. These events can be literally defined, but it must be done very carefully when framing a probability problem. From a mathematical point of view, the events are nothing more than subsets and the space of events is a Boolean algebra. Among these events, we find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events. In the experiment of rolling a die: • Event Event {3, {3, 5} (whose (whose litera literall defini definitio tionn is occurrence of 3 or 5 ) is compound because {3, 5}= {3} U {5}; • Events Events {1}, {1}, {2}, {2}, {3}, {4}, {4}, {5}, {5}, {6} are are elemen elementar tary; y; • Events {3, 5} and {4} are incompati incompatible ble or exclusive exclusive because because their intersecti intersection on is empty; that is, they they cannot occur occur simultaneously; • Events {1, {1, 2, 5} and {2, {2, 5} are nonexcl nonexclusive, usive, because because their interse intersection ction is not not empty; empty; • In the experi experiment ment of rolling rolling two dice dice one one after after another, another, the events events obtaining 3 on the first die and obtaining 5 on the second die are independent because the occurrence of the second event is not influenced by the occurrence of  the first, and vice versa. In the experiment of dealing the pocket cards in Texas Hold ’em Poker: • The event event of dealing dealing (3♣, 3♦) to a player player is an elementa elementary ry event; event; • The even eventt of of dea dealin lingg two two 3’s to a player is compound because is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦); • The events player 1 is dealt a pair of kings and player 2 is dealt a pair of kings are nonexclusive (they can both occur); • The events player 1 is dealt two connectors of hearts higher than J  and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur); • The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the occurrence of the first, while the same deck is in use). These are a few examples of gambling events, whose properties of compoundness, exclusiveness and independency are easily observable. These properties are very important in practical probability calculus. The complete mathematical model is given by the probability field attached to the experiment, which is the triple sample space—   field of events—   probability function. For any game of chance, the probability model is of the simplest type —the sample space is finite, the space of events is the set of parts of the sample space, implicitly finite, too, and the probability function is given by the definition of probability on a finite space of events:

25

Gaming mathematics

Combinations Games of chance are also good examples of combinations, permutations and arrangements, which are met at every step: combinations of cards in a player ’s hand, on the table or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on, and the like. Combinatorial calculus is an important part of  gambling probability applications. In games of chance, most of the gambling probability calculus in which we use the classical definition of probability reverts to counting combinations. The gaming events can be identified with sets, which often are sets of combinations. Thus, we can identify an event with a combination. For example, in a five draw poker game, the event at least one player holds a four of a kind formation can be identified with the set of all combinations of (xxxxy) type, where  x and  y are distinct values of cards. This set has 13C(4,4)(52-4)=624 combinations. Possible combinations are (3♠ 3♣ 3♥ 3♦ J♣) or (7♠ 7♣ 7♥ 7♦ 2♣). These can be identified with elementary events that the event to be measured consists of.

Expectation and strategy Games of chance are not n ot merely pure applications applications of probability calculus and gaming situations are not just isolated events whose numerical probability is well established through mathematical methods; they are also games whose progress is influenced by human action. In gambling, the human element has a striking character. The player is not only interested in the mathematical probability of the various gaming events, but he or she has expectations from the games while a major interaction exists. To obtain favorable results from this interaction, gamblers take into account all possible information, including statistics, to build gaming strategies. The predicted future gain or loss is called expectation or expected value and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a  fair game. The attribute  fair  refers not to the technical process of the game, but to the chance balance house (bank) – player. player. Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table —shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win. It has been mathematically proved that, in ideal conditions of randomness, no long-run regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win over the long run.

House advantage or edge Casino games generally provide a predictable long-term advantage to the casino, or "house", while offering the player the possibility of a large short-term payout. Some casino games have a skill element, where the player makes decisions; such games are called "random with a tactical element." While it is possible through skilful play to minimize the house advantage, it is extremely rare that a player has sufficient skill to completely eliminate his inherent long-term disadvantage (the house edge or house vigorish) in a casino game. Such a skill set would involve years of training, an extraordinary memory and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette. The player's disadvantage is a result of the casino not paying winning wagers according to the game's "true odds", which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6th probability of any single number appearing. However, the casino may only pay 4 times the amount wagered for a winning wager.

26

Gaming mathematics The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. (In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits.) Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38. The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%. Therefore, the house edge is 5.26%. After 10 rounds, play $1 per round, the average house profit will be 10 x $1 x 5.26% = $0.53. Of  course, it is not possible for the casino to win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round. The house edge of casino games vary greatly with the game. Keno can have house edges up to 25%, slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%. The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task. In games which have a skill element, such as Blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting), on the first hand of the shoe (the container that holds the cards). The set of the optimal plays for all possible hands is known as "basic strategy" and is highly dependent on the specific rules, and even the number of decks used. Good Blackjack and Spanish 21 games have house edges below 0.5%.

Standard deviation The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be calculated using the binomial distribution. In the binomial distribution, SD = sqrt ( npq ), where n = number of rounds played,  p = probability of winning, and q = probability of losing. The binomial distribution assumes a result of 1 unit for a win, and 0 units for a loss, rather than -1 units for a loss, which doubles the range of possible outcomes. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of  possible outcomes increases 10 fold. Therefore, SD (Roulette, even-money bet) = 2 b sqrt(npq ), where b = flat bet per round, n = number of rounds,  p = 18/38, and q = 20/38. For example, after 10 rounds at $1 per round, the standard deviation will be 2 x 1 x sqrt(10 x 18/38 x 20/38) = $3.16. After 10 rounds, the expected loss will be 10 x $1 x 5.26% = $0.53. As you can see, standard deviation is many times the magnitude of the expected loss. The range is six times the standard deviation: three above the mean, and three below. Therefore, after 10 rounds betting $1 per round, your result will be somewhere between -$0.53 - 3 x $3.16 and -$0.53 + 3 x $3.16, i.e., between -$10.00 and $8.95. (There is still a 0.1% chance that your result will exceed a $8.95 profit, and a 0.1% chance that you will lose more than $10.00.) This demonstrates how luck can be quantified; we know that if we walk into a casino and bet $5 per round for a whole night, we are not going to walk out with $500. The standard deviation for the even-money Roulette bet is the lowest out of all casinos games. Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation. As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is impossible for a gambler to win in the long term. It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

27

Gaming mathematics The volatility index (VI) is defined as the standard deviation for one round, betting one unit. Therefore, the VI for the even-money American Roulette bet is sqrt(18/38 x 20/38) = 0.499. The variance ( v) is defined as the square of  the VI. Therefore, the variance of the even-money American Roulette bet is 0.249, which is extremely low for a casino game. The variance for Blackjack is 1.2, which is still low compared to the variances of electronic gaming machines (EGMs). It is important for a casino to know both the house edge and volatility index for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the volatility index tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts. Casinos do not have in-house expertise in this field, so outsource their requirements to experts in the gaming analysis field, such as Mike Shackleford, the "Wizard of  Odds".

See also • Pro Probability • Poke Pokerr proba probabi bili lity ty in gen gener eral al • Poker Poker prob probabi ability lity (Texas (Texas hold hold 'em) 'em) • Poke Pokerr prob probab abili ility ty (Om (Omah aha) a) • Gambling • Game theory • Math Mathem emat atic icss of book bookma maki king ng

Further reading • • • •

The Mathematics of Gambling , by Edward Thorp, ISBN 0-89746-019-7 online version [1] The Theory of Gambling and Statistical Logic, Revised Edition , by Richard Epstein, ISBN 0-12-240761-X The Mathematics of Games and Gambling, Second Edition , by Edward Packel, ISBN 0-88385-646-8  Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN 973-87520-3-5 excerpts [2]

•  Luck, Logic, and White Lies: The Mathematics of Games , by Jörg Bewersdorff, ISBN 1-56881-210-8 introduction [3] .

External links • Probability Probability and and gambling gambling math math discussio discussionn from the the Wizard Wizard of Odds [4] • Application Application of probabil probability ity theory theory in games games of of chance chance [2]

References [1] [2] [3] [4]

http http:// ://ww www. w.bjmath. bjmath.com/bjmath/thorp/tog.htm com/bjmath/thorp/tog.htm http://p http://proba robabil bility ity..infarom.ro/gambling.html infarom.ro/gambling.html http http:// ://ww www. w.galois-theorie. galois-theorie.de/pdf/luck-logic-white-lies. de/pdf/luck-logic-white-lies.pdf  pdf  http://w http://wiza izardo rdofodd fodds. s.com/askthewizard/probability. com/askthewizard/probability.html html

28

Factorial number system

29

Factorial number system Numeral systems by culture Hindu-Arabic numerals

Eastern Arabic Indian family Khmer

Mongolian Thai Western Arabic

East Asian numerals

Chinese Counting rods Japanese

Korean Suzhou Vietnamese

Alphabetic numerals

Abjad Armenian Āryabhaṭa Cyrillic

Ge'ez Greek (Ionian) Hebrew

Other systems

Aegean Attic Babylonian Brahmi Egyptian Etruscan

Inuit Mayan Quipu Roman Sumerian Urnfield

List of numeral system topics Positional systems by base

Decimal (10) 1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more…

In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than n! to factorial representation, one obtains a sequence of  n digits that can be converted to a permutation of  n in a straightforward way, either using them as Lehmer code or as inversion table[1] representation; in the former case the resulting map from integers to permutations of  n lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor. [2] The term "factorial number system" is used by Knuth, [3] while the French equivalent "numérotation factorielle" is already used in 1888. [4] The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date. [5]

Factorial number system

30

Definition The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value to be multiplied by ( i − 1)! (its place value) base:

8

place value: 7! in decimal:

7

6

5

4

6!

5!

4! 3! 2! 1! 0!

5040 720 120 24 6

3 2

2

1

1

1

So the rightmost digit is always 0, the second can be 0 or 1, the third 0, 1 or 2, and so on. The factorial number system is sometimes defined with the rightmost digit omitted, because it is always zero (sequence A007623 [6] in OEIS). In this article a factorial number representation will be flagged by a subscript "!", so for instance 341010 ! stands for 364514031201, whose value is ((((3×5 + 4)×4 + 1)×3 + 0)×2 + 1)×1 + 0 = 463 10. General properties of mixed radix number systems apply to the factorial number system as well. For instance, one can convert a number into factorial representation producing digits from right to left, by repeatedly dividing the number by the place values (1, 2, 3, ...), taking the remainder as digits, and continuing with the integer quotient, until this quotient becomes 0. One could in principle extend the system to deal with fractional numbers by choosing base values for the positions after the "decimal" point, but the natural extension by values 0, −1, −2, ... is not an option. The symmetric choice of base values 1, 2, 3, ... after the point would be possible, with corresponding place values 1  ⁄ n!, but it is not distinguished by an particular mathematical properties (except that the number e takes the form 10.011111...).

Examples Here are the first twenty-four numbers, counting from zero, in factorial representation: deci decima mall fact factor oria iall

0

0!

1

10!

2

100!

3

110!

4

200!

5

210!

6

1000!

7

1010!

8

1100!

9

1110!

10

1200!

11

1210!

12

2000!

13

2010!

14

2100!

15

2110!

16

2200!

Factorial number system

31 17

2210!

18

3000!

19

3010!

20

3100!

21

3110!

22

3200!

23

3210!

For another example, the biggest number that could be represented with six digits would be 543210 ! which equals 719 in decimal: 5×5! + 4×4! + 3×3! + 2×2! + 1×1! + 0×0!. Clearly the next factorial number representation after 543210 ! is 1000000! which designates 6! = 720 10, the place value for the radix-7 digit. So the previous number, and its summed out expression above, is equal to: 6! − 1. The factorial number system provides a unique representation for each natural number, with the given restriction on the "digits" used. No number can be represented in more than one way because the sum of consecutive factorials multiplied by their index is always the next factorial minus one:

This can be easily proved with mathematical induction. However, when using arabic numerals to write the digits (and not including the subscripts as in the above examples), their simple concatenation becomes ambiguous for numbers having a "digit" bigger than 9. The smallest such example is the number 10 × 10! = 3628800010, which may be written A0000000000 !, but not 100000000000 ! which denotes 11!=3991680010. Thus using letters A – Z to denote digits 10, ..., 35 as in other base-N make the largest representable number 36! − 1=37199332678990121746799944815083519999999910. For arbitrarily larger numbers one has to choose a base for representing individual digits, say decimal, and provide a separating mark between them (for instance by subscripting each digit by its base, also given in decimal). In fact the factorial number system itself  is not truly a numeral system in the sense of providing a representation for all natural numbers using only a finite alphabet of symbols.

Permutations There is a natural mapping between the integers 0, ..., n! − 1 (or equivalently the numbers with n digits in factorial representation) and permutations of  n elements in lexicographical order, when the integers are expressed in factoradic form. This mapping has been termed the Lehmer code (or inversion table). For example, with n = 3, such a mapping is

Factorial number system

32

deci decima mall fact factor oria iall perm permut utat atio ionn 010

000!

(0,1,2)

110

010!

(0,2,1)

210

100!

(1,0,2)

310

110!

(1,2,0)

410

200!

(2,0,1)

510

210!

(2,1,0)

The leftmost factoradic digit 0, 1, or 2 is chosen as the first permutation digit from the ordered list (0,1,2) and is removed from the list. Think of this new list as zero indexed and each successive digit dictates which of the remaining elements is to be chosen. If the second factoradic digit is "0" then the first element of the list is selected for the second permutation digit and is then removed from the list. Similarly if the second factoradic digit is "1", the second is selected and then removed. The final factoradic digit is always "0", and since the list now contains only one element it is selected as the last permutation digit. The process may become clearer with a longer example. For example, here is how the digits in the factoradic 4041000! (equal to 298210) pick out the digits in (4,0,6,2,1,3,5), the 2982nd permutation of the numbers 0 through 6. 4041000 factoradic: factorad ic:

!

→ (4,0,6,2,1,3,5)

4

0

4

1

0

0

0

|

|

|

|

|

|

|

!

(0,1,2,3,4,5,6) -> (0,1,2,3,5,6) -> (1,2,3,5,6) -> (1,2,3,5) -> (1,3,5) -> (3,5) -> (5) | permutation:(4,

|

|

|

|

|

|

0,

6,

2,

1,

3,

5)

A natural index for the group direct product of two permutation groups is the concatenation of two factoradic numbers, with two subscript "!"s. concatenated decimal 0 1

10 10

factoradics

permutation pair

000 000

((0,1,2),(0,1,2))

000 010

((0,1,2),(0,2,1))

! !

! !

... 5 6 7

10 10 10

000 210

((0,1,2),(2,1,0))

010 000

((0,2,1),(0,1,2))

010 010

((0,2,1),(0,2,1))

! ! !

! ! !

... 22

10

110 200 !

!

((1,2,0),(2,0,1))

... 34

210 200

((2,1,0),(2,0,1))

35

210 210

((2,1,0),(2,1,0))

10 10

! !

! !

Factorial number system

External links • Mantaci, Mantaci, Roberto; Roberto; Rakotondra Rakotondrajao, jao, Fanja Fanja (2001), "A permutatio permutationn representatio representationn that knows what what “Eulerian” [7] means" (PDF), Discrete Mathematics and Theoretical Computer Science 4: 101 – 108. 108. • Arnd Arndt, t, Jör Jörgg (Mar (March ch 5, 5, 2009 2009). ). Algorithms for Programmers: Ideas and source code (draft) [8]. pp. 224 – 234. 234.

See also • Combinatori Combinatorial al number number system (also called called combinadi combinadics) cs) • Factorial

External links • A Lehm Lehmer er cod codee cal calcu cula lato torr [9] Note that their permutation digits start from 1, so mentally reduce all permutation digits by one to get results equivalent to the ones on this page

References [1] Knuth, D. D. E. (1973), "Volume "Volume 3: Sorting Sorting and and Searching", Searching", The Art of Computer Programming, Addison-Wesley, pp. 12, ISBN 0-201-89685-0 [2] Cantor, Cantor, G. (1869 (1869), ), Zeitschrift für Mathematik und Physik , 14. [3] Knuth, D. E. (1997), (1997), "Volume "Volume 2: Seminumerical Seminumerical Algorithms Algorithms", ", The Art of Computer Programming (3rd ed.), Addison-Wesley, pp. 192, ISBN 0-201-89684-2. [4] Laisant, Charles-Ange Charles-Ange (1888), "Sur "Sur la numération factorielle, application aux permutations" (http://www.numdam. (http://www.numdam.org/  org/  item?id=BSMF_1888__16__176_0) (in French), Bulletin de la Société Mathématique de France 16: 176 – 183, 183, . [5] The term "factoradic "factoradic"" is apparently apparently introduced in McCaffrey, McCaffrey, James James (2003), Using Permutations in .NET for Improved Systems Security (http://msdn2.microsoft. (http://msdn2. microsoft.com/en-us/library/aa302371. com/en-us/library/aa302371. aspx), Microsoft Developer Network, , which claims to present a previously unpublished algorithm to generate permutations using a construction called the "factoradic", apparently ignorant of previous work on the factorial number system. [6] http http:// ://en en..wikipedia.org/wiki/Oeis%3Aa007623 wikipedia. org/wiki/Oeis%3Aa007623 [7] http http:// ://ww www. w.dmtcs. dmtcs.org/volumes/abstracts/pdfpapers/dm040203. org/volumes/abstracts/pdfpapers/dm040203. pdf  [8] http http:// ://ww www. w. jjj.de/fxt/#fxtbook  jjj.de/fxt/#fxtbook [9] http://w http://wwwww-ang ang..kfunigraz.ac.at/~fripert/fga/k1lehm. kfunigraz.ac.at/~fripert/fga/k1lehm. html

33

Numeral system

34

Numeral system Numeral systems by culture Hindu-Arabic numerals

Eastern Arabic Indian family Khmer

Mongolian Thai Western Arabic

East Asian numerals

Chinese Counting rods Japanese

Korean Suzhou Vietnamese

Alphabetic numerals

Abjad Armenian Āryabhaṭa Cyrillic

Ge'ez Greek (Ionian) Hebrew

Other systems

Aegean Attic Babylonian Brahmi Egyptian Etruscan

Inuit Mayan Quipu Roman Sumerian Urnfield

List of numeral system topics Positional systems by base

Decimal (10) 1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more…

A numeral system (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner. It can be seen as the context that allows the numerals "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. Ideally, a numeral system will: • Represent Represent a useful useful set of numbers numbers (e.g. (e.g. all all integers, integers, or rational rational numbers) numbers) • Give every every number number represented represented a unique unique representa representation tion (or at least least a standard standard representati representation) on) • Reflect Reflect the algebra algebraic ic and arithmetic arithmetic structu structure re of the the numbers. numbers. For example, the the usual decimal decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits. However, when decimal representation is used for the rational or real numbers, the representation may not be unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except for some scientific contexts where greater precision is implied by the trailing zero. Numeral systems are sometimes called number systems, but that name is misleading, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of   p-adic numbers, etc. Such systems are not the topic of this article.

Numeral system

Types of numeral systems The most commonly used system of numerals is known as Hindu-Arabic numerals, and two Indian mathematicians are credited with developing them. Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero. [1] The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////. Tally marks represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is commonly used in data compression, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral. The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + - - /// without any need for zero. This is called sign-value notation. The ancient Egyptian numeral system was of this type, and the Roman numeral system was a modification of this idea. More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D/ for the number 304. The number system of the English language is of this type ("three hundred [and] four"), as are those of other spoken languages, regardless of what written systems they have adopted. However many languages use mixtures of bases, and other features, for instance 79 in French is soixante dix-neuf  (60+10+9) and in Welsh is   pedwar ar bymtheg a thrigain (4+(5+10)+(3 x 20)) or (somewhat archaic)  pedwar ugain namyn un (4 x 20 - 1) More elegant is a   positional system, also known as place-value notation. Again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system. Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only 10 different symbols (assuming that it uses base 10). The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals. In certain areas of computer science, a modified base- k  positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by an empty string. s tring. This establishes a bijection between the set of  all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-k numeration is also called k -adic -adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

35

Numeral system

36

Positional systems in detail In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b. For example, in the decimal system (base 10), the numeral 4327 means ( 4×103) + (3×102) + (2×101) + (7×100), noting that 100 = 1. In general, if  b is the base, we write a number in the numeral system of base b by expressing it in the form anbn + an − 1bn − 1 + an − 2bn − 2 + ... + a0b0 and writing the enumerated digits anan − 1an − 2 ... a0 in descending order. The digits are natural numbers between 0 and b − 1, inclusive. If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number base. Unless specified by context, numbers without subscript are considered to be decimal. By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2 1 + 0×20 + 1×2−1 + 1×2−2 = 2.75. In general, numbers in the base b system are of the form:

The numbers bk  and b−k  are the weights of the corresponding digits. The   position k is the logarithm of the corresponding weight w, that is . The highest used position is close to the order of  magnitude of the number. The number of tally marks required in the unary numeral system for describing the weight  would have been w. In the positional system the number of digits required to describe it is only , for . E.g. to describe the weight 1000 then four digits are needed since

. The number of digits

required to describe the position is

(in positions 1, 10, 100... only for simplicity in

the decimal example). Position

3

2

1

0

-1

-2

...

Weight

...

Digit

...

Decimal example weight 1000 100 10 Decimal example digit

4

3

2

1

0.1

0.01 ...

7

0

0

...

Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3 10 = 0.0100110011001...2). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...10 can be written down as the unperiodic 11.001001000011111... 2.

Putting overscores, n, or dots, •n, above the common digits is a convention used to represent repeating rational expansions. Thus: 14/11 = 1.272727272727... = 1.27 or 321.3217878787878... = 321.321•7•8 .

If  b = p is a prime number, one can define base- p numerals whose expansion to the left never stops; these are called the p-adic numbers.

Numeral system

37

Generalized variable-length integers More general is using a notation (here written little-endian) like

for

, etc.

This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of  arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a – z and 0 – 9, 9, representing 0 – 25 25 and 26 – 35 35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b – 9 (1 – 35), 35), therefore the weight b1 is 35 instead of 36. Suppose the threshold values for the second and third digits are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence: a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc. Unlike a regular based numeral system, there are numbers like 9b where 9 and b each represents 35; yet the representation is unique because ac and aca are not allowed  – the a would terminate the number. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of  numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are non-zero.

See also • • • • • • • • • • •

Baby Babylo loni nian an nume numera rals ls – a sexagesimal (base-60) system Comp Comput uter er num numbe beri ring ng form format atss Gold Golden en rati ratioo base base List List of num numer eral al sys syste tem m topi topics cs Maya Maya nume numera rals ls – a base-20 system N-ary Number na names Quipu Recu Recurr rrin ingg dec decim imal al Resi Residu duee numb number er syste system m Subt Subtra ract ctiv ivee nota notati tion on

References [1]  Hindu Arabic Numerals by David Eugene Smith Google Books) (http://books.google. (http://books.google.com/books?id=wEw6AAAAMAAJ&dq=eugene+ com/books?id=wEw6AAAAMAAJ&dq=eugene+ smith+hindu&printsec=frontcover&source=bl&ots=fEff_4LbmT&sig=IpstDJbzWhBaElmA5AvlZ7Ps2lY&hl=en&sa=X& oi=book_result&resnum=1&ct=result)

• Geor George gess Ifr Ifrah ah.. The Universal History of Numbers : From Prehistory to the Invention of the Computer , Wiley, 1999. ISBN 0-471-37568-3. • D. Knuth. The Art of Computer Programming . Volume 2, 3rd Ed. Addison-Wesley. pp. 194 – 213, 213, "Positional Number Systems". • A. L. Kroeber Kroeber (Alfred (Alfred Louis Kroeber) Kroeber) (1876 (1876 - 1960), Handbook Handbook of the the Indians Indians of California, California, Bulletin Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919) • J.P. J.P. Mall Mallor oryy and and D.Q. D.Q. Ada Adams ms,, Encyclopedia of Indo-European Culture , Fitzroy Dearborn Publishers, London and Chicago, 1997. • Hans Hans J. Nisse Nissen, n, P. Damero Damerow, w, R. R. Englun Englund, d, Archaic Bookkeeping , University of Chicago Press, 1993, ISBN 0-226-58659-6.

Numeral system • Deni Denise se Schma Schmand ndtt-Be Besse ssera rat, t, How Writing Came About , University of Texas Press, 1992, ISBN 0-292-77704-3. • Clau Claudi diaa Zasl Zaslav avsk sky, y, Africa Counts: Number and Pattern in African Cultures , Lawrence Hill Books, 1999, ISBN 1-55652-350-5.

External links • Numerical Numerical Mechan Mechanisms isms and Children's Children's Concep Conceptt of Numbers Numbers (http://web (http://web.. media.mit.edu/~stefanm/society/  media.mit.edu/~stefanm/society/  som_final.html) som_final.html) • Software Software for converti converting ng from one one numeral numeral system system to another another (http://billpose (http://billposer. r.org/Software/libuninum. org/Software/libuninum.html) html)

List of numeral system topics This is a list of numeral system topics and "numeric representations". It does not systematically list computer formats for storing numbers, see also: computer numbering formats and number names.

Arranged by base • Radix, Radix, radix radix point, point, mixed radix, base (mathematics (mathematics)) • Unar Unaryy nume numera rall syste system m (base (base 1) • • • • • • • •

• Tally ma marks Binary Binary numera numerall syste system m (bas (basee 2) 2) Negati Negative ve base base numera numerall syste system m (base (base −2) Ternary Ternary numer numeral al system system numer numeral al system system (base (base 3) Balanc Balanced ed terna ternary ry numer numeral al syste system m (base (base 3) Negati Negative ve base base numera numerall syste system m (base (base −3) Quater Quaternar naryy numer numeral al syst system em (bas (basee 4) Quater Quater-im -imagi aginar naryy base base (base (base 2√−1) Quinar Quinaryy nume numeral ral system system (base (base 5)

• • • • •

• Pent Pentim imal al sys syste tem m Senary Senary numera numerall syste system m (bas (basee 6) Septen Septenary ary numera numerall syste system m (base (base 7) Octa Octall nume numera rall syste system m (bas (basee 8) Nonary Nonary (nov (novena enary) ry) nume numeral ral system system (base (base 9) Decima Decimall (denar (denary) y) numer numeral al syste system m (base (base 10) 10)

• • • • •

• Bi-q Bi-qui uina nary ry cod coded ed dec decim imal al Negati Negative ve base base numera numerall syste system m (base (base −10) Duodec Duodecima imall (dozena (dozenal) l) numera numerall system system (base 12) 12) Hexade Hexadecim cimal al numer numeral al syste system m (base (base 16) 16) Vigesi Vigesimal mal numera numerall system system (base (base 20) 20) Sexage Sexagesima simall numer numeral al syst system em (bas (basee 60) 60)

38

List of numeral system topics

Arranged by culture • • • •

Austra Australia liann Abori Aborigin ginal al enum enumera eratio tionn Arme Armeni nian an nume numera rals ls Baby Babylo loni nian an nume numera rals ls Chin Chines esee nume numera rals ls

• • • • •

• Coun Counti tinng rods rods Colo Colomb mbia iann nume numera rals ls Aege Aegean an numb number erss Gree Greekk num numer eral alss Hebr Hebreew nume numera rals ls Hindu – Arabic Arabic numeral system

• • • • • • •

• Arab Arabic ic num numer eral alss • Indi Indian an num numeerals rals • Thai Thai nume numera rals ls Japa Japane nese se nume numera rals ls Kore Koreaan nume numera rals ls Maya Maya nume numera rals ls Moks Moksha ha num numer eral alss Prehi Prehisto stori ricc nume numera rals ls Roma Romann num numer eraals Wels Welshh num numeerals rals

Other • • • • • • •

Algorism Good Goodst stei ein' n'ss theor theorem em Myriad Non-sta Non-standa ndard rd positi positiona onall numera numerall systems systems Quipu Tally st stick Tally mark

39

Article Sources and Contributors

Article Sources and Contributors http://en.wikipedia.org/w/index.php?oldid=366119802 p?oldid=366119802 Contributors: A D Monroe III, Aca d Ronin, Aces&8s, Adashiel, Administrationers, Admoose, Afil, AgentPeppermint, Source: http://en.wikipedia.org/w/index.ph Albrecht, Alexius08, Alphachimp, Andrewpmk, Andycjp, Angry Sun, Another mutant, Apewty111, ApostleJoe, AquafireGal, Arenarax, Argon233, Arichnad, Augustus Rookwood, BaShildy, Bahar101, Batsabrina, Behack, Bjones, Bkonrad, Blooddraken, Bmrbarre, BoNoMoJo (old), Bobo192, Bongwarrior, Brenont, Bubbha, CambridgeBayWeather, Can't sleep, clown will eat me, Capricorn42, Captwheeler, CarbonCopy, Carlo V. Sexron, Carlossuarez46, Carterbrumm, Catherineyronwode, Cautious, Chris the speller, Chronocore, Ckatz, Clam0p, ClubOranje, Cometstyles, Crownie601, Crucis, Cuttycuttiercuttiest, Cyberdrummer, DMG413, Danielsilliman, Dasani, Dashboard923, David Shay, Dee lkar, Deltadot, Dgmendez, Dhammapal, Dicklyon, Dino, Discospinster, DocPsych, DoctorWho42, Dodiad, Dp, Drini, Drmagic, ESkog, EamonnPKeane, Ebyabe, Edward blue, Eelke, Eggsyntax, Electronic.mayhem, ElfQrin, Eliyak, Em-jay-es, Emilydenton, Emurphy42, Epbr123, Eric outdoors, Esprit15d, Eteq, Ettiesniffs, Everyking, Everyoneandeveryone, Evil Monkey, Evolutionnz, Falcon8765, Farij, Fasach Nua, Fconaway, Firien, Former user 2, Funandtrvl, GB fan, GeneMosher, GenkiNeko, George100, Giggy, Gil-Galad, Gilabrand, GorillazFanAdam, GraemeL, Gregbard, Grick, Gwib, Hadal, Hdt83, Heqs, Hgchan888, Hibernian, Hide&Reason, Ihcoyc, Iridescent, Ithizar, Izehar, J.delanoy, JHFTC, JHunterJ, JackLumber, Jacklonergan, Jeff G., Jemesouviens32, Jennavecia, Jeong Hwa-Yong, Jimmyeightysix, Jjones909, Jjyet, Jmlk17, John254, JohnCD, Johnfn, Joyrika, Jphang75, Juicedpixels, Junyor, Jusdafax, Kamranhk, Katieh5584, Kenheut, Keolah, Kgasso, Khym Chanur, Killer3000ad, Kku, Knowpedia, KrakatoaKatie, Kubigula, Kungfuadam, Kuru, Kyz, LOL, Lacrimosus, Lamcs, LeaveSleaves, LedgendGamer, Ledzelda, Lenoxus, Leonard G., Leotardo, Liftarn, Lindberg G Williams Jr, Logarithm88, LokalLuzer, Lterra, Luckisnolady, LuckyStar, LukeSurl, LulzyLulz, Luna Santin, Mac Davis, MafiaCapo, Mandarax, Marasama, MarcoPalacios, Marek69, Mark T, Marshall Williams2, Marshamarsha123, Mattbrundage, Mboverload, McNoddy, Mdebets, Meaghan, MegX, Michaelritchie200, Michal Nebyla, Mickeyg13, Mike Rosoft, Minimac, Miss Sabre, Misza13, Momoricks, Moondoll, Mschlindwein, Mushin, NHRHS2010, Nameologist, NawlinWiki, NealIRC, Neelix, NellieBly, Nicholasink, Nicolae Coman, Noon, Nutiketaiel, ONEder Boy, Occono, Ohnoitsjamie, Open2universe, Orijok, Oxymoron83, Oytun 73, Pamporoff, Pa rker007, Pearle, Pedro, Petertjuuh26, Peterwhy, Pfalstad, Pigman, Pikiwedian, Pincushin, PrestonH, Psiph, Psychonaut, Qtac, Qz, RJFerret, RandomStringOfCharacters, Ray-Ginsay, Reinyday, RexNL, Richard Weil, Richard001, Richrich27, Rjwilmsi, Rogerious, RoyBoy, Rrmsjp, Rusty2005, SRauz, SWAdair, Salanth, Sam Hocevar, Samaritan, Samir, Samw, SanguineF, Sannse, Sciurinæ, Scoutersig, Sexford, ShelfSkewed, Sherbrooke, Shereth, Shirik, Sinisterscrawl, Sirkar118, Skiboarder6730, Skjolly, Snigbrook, SnowFire, Sommers, Spencerk, Startstop123, Stephenb, Steve3849, Sturm55, Suffusion of Yellow, Swatjester, T-Money92, TastyPoutine, Tchoutoye, Tealwisp, Ter308, Texture, The Iconoclast, The Wookieepedian, TheSuave, TheTrojanHought, Themagicalpixieelf, Themfromspace, Themissinglint, Tidus the BlitzStar, Tiptoety, Tom harrison, Tommy2010, Tony Sidaway, Treisijs, Trevor MacInnis, True P agan Warrior, TrunksWP, Tsemii, Turian, Tuzapicabit, Tylerdmace, Uhgoidhgfhg, Unclekirk, User27091, UtherSRG, VIP26, Versus22, Vicarious, Vipinhari, Wack'd, Wayward, Weirdoactor, West wikipedia, WholemealBaphomet, Wik, Windward1, Wiredabc, Woodshed, X!, Xiahou, XxTimberlakexx, Ysangkok, Zackcordle, Zanter, Zeality, Zondor, 755 anonymous edits

Luck

http://en.wikipedia.org/w/index.php?oldid=366804592 dex.php?oldid=366804592 Contributors: 159753, 2005, 9Nak, A bit iffy, A little insignificant, Accurizer, Adam Conover, Aervanath, Ahda, Source: http://en.wikipedia.org/w/in Ahoerstemeier, Aiman abmajid, Alan Liefting, Ale jrb, Andrevan, Andrewpmk, Andrzej P. Wozniak, AnonUser, Antandrus, Anwar saadat, Arenarax, Avala, AxG, B.S. Lawrence, BD2412, BKfi, Bairam, Barneyboo, Barticus88, Bayerischermann, Beeswaxcandle, Belasted, Benphang, Blackmagicfish, Blah123454321, Bluerasberry, Bluewave, Bookandcoffee, Borgx, BrettAllen, BriGuy20, Brianga, Brice Hutfles, Bumm13, ByeByeBaby, CJ, CR85747, Calm, Caltas, CambridgeBayWeather, Can't sleep, clown will eat me, Capricorn42, Carefair, CaribDigita, Carton828, Cassowary, Cfradut, Chrisd87, Cncs wikipedia, Colantim, CrashMex, Crd721, Cripipper, CroatiaShoes, CryptoDerk, Cryptoid, Cureden, DO'Neil, DSRH, DV8 2XL, DanTilkin, Dario20009, Dark spartan082, DarkDespair5, DarthVader, Dave6, DavidA, Davidparry, Dbenbenn, Dejvas, Delirium, Demiester, Deon Steyn, DerHexer, DeuceBigalowMD, Dhodges, Dhollm, Digana, Djegan, Dppowell, Dromedary, Dryazan, Dshin, Dubloon, Ducks Are stupid, Dylan Stafne, EamonnPKeane, Ebricca, Ehrenkater, Eivind F Øyangen, El C, Elfguy, Elwood00, Emijrp, Eminem depicts a clown with blazing hair., EugeneZelenko, Ev, Everyking, Extrahitz, Feedmecereal, Fg2, FishSpeaker, Flrn, Flyguy649, Fonzy, FoxLad, Funfb, Furrykef, Gary D, Gaspel223, GeneralBelly, Geoff43230, Gigsinho, Gintyfrench, GraemeL, Granzo, Grassfire, Greenboxed, Gregory Arkadin, Grogo21, Ground Zero, Guy in rollerblades, Guyzero, Gypsum Miner, Haakon, Haham hanuka, Hardyplants, Haricotvert, Haukacik, Henning Makholm, Heresthecasey, Heron, Hiroe, Hristodulo, IPSkaltsas, Idleguy, Iloveads47, Infocraze, Invader chris, Ionware, Izmaelt, JForget, JFreeman, JPG-GR, Jack Cox, Jack.l.stamp, Janke, Jarmo Gombos, Jasonr lau, Jiddisch, JimmySand9, Jmlk17, John Riemann Soong, John Smythe, John wesley, Joost 99, JorgeGG, Jouster, Joy, Jsonitsac, Jwmcleod, JzG, Kanyewes 05, Keith Lehwald, Kerub, Keycard, Kirjtc2, Koavf, Kungfuadam, Kvenner, La goutte de pluie, Lacrimosus, Lalolanda, Lambton, Lars Washington, Lazyern, Lazzy newtt, Lcmortensen, LittleOldMe, LiveAgain, Locnguyun, Lottery Universe, Lottery01, LottsoLuck, Lova Falk, Love Krittaya, LovesGod, Loveyourcar, Lozeldafan, LukeD, LukeSurl, Lukereiser, Lundleo, Lusanaherandraton, Macarism, MachineFist, Mackeriv, Marc van Leeuwen, MarcMacé, Marchije, Mardochaios, Martin451, MartinRe, Mbc362, Mbxp, McDogm, Mcleanclan, Micahbrwn, MikeGermano, Misterioso del 97, Mks004, Moncrief, Monedula, Monkeyman, Moogle10000, Mordien, Mothmolevna, Mshahidnawaz9, Mtreinik, NTK, Nadav1, Nadiral, Nakadai, Nakakapagpabagabag, Nanami Kamimura, Natedogg60, NawlinWiki, New Thought, Nic bor, Night Tracks, Noirum, Novacatz, Nowa, Nsaa, Nurg, Olivier, Onlytwice, Oxymoron83, P The D, Paine Ellsworth, Parpaluck, Patrick, Paul Carpenter, PaulMottley, Pcb21, Pcpp, Persian Poet Gal, Pfrancing, Phgao, Piedude22, PikDig, Piledhigheranddeeper, Polly, Poor Yorick, RA0808, RHaworth, RandomStringOfCharacters, Rapido, Rchamberlain, Readparse, RedWolf, Redrocket, Rfc1394, Rforeman, Rich Janis, Rjwilmsi, Roaming, Rob.derosa, Robvhoorn, Ronshelley, Rray, Rrburke, Rufus843, SDC, SE KinG, SGBailey, Saaga, Sahmeditor, Samw, Saric, Saxo Grammaticus, Sceptre, SchfiftyThree, ScotchMB, Scroch, Secfan, Shanes, Sharkford, Shell Kinney, Sl, Smalljim, SmartGuy, Smilesfozwood, Smiley22, Snappy, Sommers, Sonett72, Sortior, Soulpatrol, Speednet, Sport woman, Sry85, Stalmannen, Starghost, Static Universe, Steviej815, StuRat, Sunindia14, Taintain, TakuyaMurata, Tassedethe, Tbhotch, Techsmith, Texchen, The undertow, TheGrza, TheLotter, Thelmadatter, Themepark, Thumperward, Tide rolls, Tigeron, TimBentley, Timwi, Tkondaks, Toh, Tohd8BohaithuGh1, Tony1, TrbleClef, Trusilver, TruthIsNeverTooHorrible, Twas Now, US N1977, Unigolyn, Unkx80, Utility Monster, Valodzka, Vegaswikian, Verne Equinox, Violetriga, Wcudmore, Wdfarmer, Wetman, WikiNazi, Wikipromedia, Willhester, Wimt, Wolf530, Wperdue, Xanthar, Xelaxa, Xx236, Zaian, Zanimum, Zjc263, Ævar Arnfjörð Bjarmason, రవిచంద్ చంద్ ర, 706 anonymous edits

Lottery

http://en.wikipedia.org/w/index.php?oldid=358361663 ex.php?oldid=358361663 Contributors: Eraoul, Giftlite, JamesDmccaffrey, Jwmcleod, Lantonov, Lasloo, LeeHunter, Marc Source: http://en.wikipedia.org/w/ind van Leeuwen, Michael Hardy, Oleg Alexandrov, Salix alba, Semifinalist, The Anome, Txen, Whpq, Zaslav, 11 anonymous edits Combinatorial number system

Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?oldid=358207180 dex.php?oldid=358207180 Contributors: 2005, AirdishStraus, Anwhite, Arthur Rubin, Dale Arnett, Doug Bell, Geometry guy, Iezegrim, Infarom, Ixfd64, Kevin Forsyth, Larrymcp, Melcombe, Michael Hardy, Norbiton, ONEder Boy, Oasisbob, Packel, PhGustaf, PrimeHunter, Rray, Tirkfl, Trovatore, 33 anonymous edits Gaming mathematics

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Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?oldid=367087670 dex.php?oldid=367087670 Contributors: 16@r, 4pq1injbok, 64.26.98.xxx, 75th Trombone, A. B., Abednigo, Adam majewski, Adhemar, Aitias, Altenmann, Andrea.gf, AndrewKepert, Andrewpmk, AnonMoos, Anoopan, Arthur R ubin, Atlant, AxelBoldt, AySz88, BD2412, Bduke, Bfinn, B illcito, Blue520, Blumpkin, Bobak, Bogdangiusca, Bornintheguz, Britcom, Bryan Derksen, Buck Mulligan, Carifio24, Cbdorsett, Celada, Ceranthor, Chmod007, Ckatz, Conversion script, Cowenby, Ctachme, Cyan, Cyp, DV8 2XL, Dan 9111, DanP, Daniel Lawrence, Date20070309, DavidCBryant, DavidCary, Davilla, Dbachmann, Dcoetzee, Deeptrivia, Denelson83, Deor, Discospinster, Dogcow, Dreadstar, Dysprosia, EMU CPA, EamonnPKeane, Easytoremember, Eclecticology, Eequor, Efini, Ehrenkater, Epbr123, Eras-mus, Eric Kvaalen, EugeneZelenko, Evercat, Ezra Katz, Fgrosshans, FilipeS, Fred Bauder, Fredrik, Fryed-peach, Galoubet, Garyvdm, Gavroche42, Geekygator, Geoffrey, Geometry guy, Giftlite, Glenn, Glenn L, Goochelaar, Gpvos, Graham87, Greensburger, Gus Polly, Haeleth, Hardpack, Henning Makholm, Henrygb, IL-Kuma, Ica irns, Infrogmation, J.Rohrer, JYolkowski, Jagged 85, Jaredwf, Jerz y, Jesset77, Jiddisch, Jim.henderson, Jimp, Jleedev, JoanQ, JohnBlackburne, JohnOwens, Joyous!, Jpbowen, Jshadias, Julian Diamond, Karl Palmen, KnowledgeOfSelf, Koavf, Krajcsi, Krich, LachlanA, Lacrimosus, Le Anh-Huy, Lemmiwinks2, Lenoxus, LiDaobing, LilHelpa, Looxix, Lucinos, Lumos3, LusciniaMegarhynchos, M a s, MacGyverMagic, Mad Cat, Maelnuneb, Mark R Johnson, Maximaximax, Mbarbier, McGeddon, Mendalus, Michael Hardy, Mightyeldude, Mild Bill Hiccup, Milogardner, Mlpkr, Mm1972, Mo-Al, Mr Adequate, Myanw, Mzajac, Najro, Nascigl, Nat Krause, Nihiltres, Nikola Smolenski, Noe, Noisy, OMGsplosion, Octahedron80, Oleg Alexandrov, OrgasGirl, Palapala, Patrick, Patstuart, Paul August, Paul Stansifer, Paxsimius, Peak, P edalist, Pengo, Philip Trueman, Phoenix-forgotten, Pianoplonkers, Pinethicket, Pinnerup, Pizza Puzzle, Python eggs, Qaz, Quanstro, Qwfp, R.e.s., Raj2004, Rajah, Ran, Rashaani, Rettetast, Rich Farmbrough, Robertolyra, Rorro, Ruud Koot, Ruwanraj, SDaniel, SGBailey, Sam Hocevar, Shawnhath, Shellreef, Shorne, Silverxxx, Sligocki, SomeHuman, Sonjaaa, Spoon!, Srulikbd, Stephan Leeds, Stevan White, Sunborn, Suruena, TakuyaMurata, Template namespace initialisation script, The Thing That Should Not Be, Theinfo, Tim Andrews, Timwi, Tristanb, Trojo, Trumpet7, Tuomas Toivonen, Turbojet, Twanderson, Uriyan, Vanjagenije, Velho, Viznut, Wapcaplet, Wereon, Wiki alf, Wiwaxia, XJamRastafire, Yugyug, Zachorious, Zundark, 360 anonymous edits

Numeral system

Source: http://en.wikipedia.org/w/index.php http://en.wikipedia.org/w/index.php?oldid=351648747 ?oldid=351648747 Contributors: 75th Trombone, AnonMoos, Aranel, Charles Matthews, Crazymadlover, D6, Deeptrivia, Ejilgun, FilipeS, Fplay, Funandtrvl, Hipocrite, Ipi31415, Jagged 85, Kar l Palmen, Michael Hardy, Paul Martin, Portalian, Raoul NK, Rpchase, TAKASUGI Shinji, The Transhumanist, ZeroOne, 7 anonymous edits List of numeral system topics

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Image Sources, Licenses and Contributors

41

Image Sources, Licenses and Contributors Image:Four-leaf clover.jpg

Source: http://en.wikipedia.org/w/ind http://en.wikipedia.org/w/index.php?title=File:Four-leaf_clover.jpg ex.php?title=File:Four-leaf_clover.jpg  License: GNU Free Documentation License Contributors: User:Phyzome

Source: http://en.wikipedia.org/w/index http://en.wikipedia.org/w/index.php?title=File:1926WhyBeUnlucky.jpg .php?title=File:1926WhyBeUnlucky.jpg  License: Public Domain Contributors: anon work. Magazine "Art and Be auty Magazine" was published by "the King Publishing Co., Wilmington, Del.". Image:1926WhyBeUnlucky.jpg

Source: http://en.wikipedia.org/w/index.php?title=File:Maneki_neko_with_7_Lucky_Gods_by_OiMax_in_ http://en.wikipedia.org/w/index.php?title=File:Maneki_neko_with_ 7_Lucky_Gods_by_OiMax_in_Asakusa,_Tokyo.jpg Asakusa,_Tokyo.jpg  License: Creative Commons Attribution 2.0 Contributors: FlickreviewR, Opponent Image:Maneki neko with 7 Lucky Gods by OiMax in Asakusa, Tokyo.jpg

Image:National Lottery play here! sign.jpg

Source: http://en.wikipedia.org/w/index.ph http://en.wikipedia.org/w/index.php?title=File:National_Lottery_pl p?title=File:National_Lottery_play_here!_sign.jpg ay_here!_sign.jpg  License: unknown Contributors:

User:BotMultichillT Image:NationalLotteryBldgMexico.JPG Image:English Lottery 1567 001.jpg

Source: http://en.wikipedia.org/w/index http://en.wikipedia.org/w/index.php?title=File:NationalLotteryBldgM .php?title=File:NationalLotteryBldgMexico.JPG exico.JPG  License: Public Domain Contributors: User:Thelmadatter

http://en.wikipedia.org/w/index.php?title=File:English_Lott .php?title=File:English_Lottery_1567_001.jpg ery_1567_001.jpg  License: GNU Free Documentation License Contributors: Source: http://en.wikipedia.org/w/index

User:Ronshelley File:1814 English State Lottery Ticket.jpg

Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?title=File:1814_Englis dex.php?title=File:1814_English_State_Lottery_Ticket.jpg h_State_Lottery_Ticket.jpg  License: GNU Free Documentation License

Contributors: User:Ronshelley Source: http://en.wikipedia.org/w/index.php?title=File:Lottery_ticket_-_Queen's_College_Lottery,_New-Brunswick,_New_Jersey,_USA,_1814.jpg http://en.wikipedia.org/w/index.php?title=File:Lottery_ticket_-_Queen's_Col lege_Lottery,_New-Brunswick,_New_Jersey,_USA,_1814.jpg  License: Public Domain Contributors: State of New Jersey, USA Image:Lottery ticket - Queen's College Lottery, New-Brunswick, New Jersey, USA, 1814.jpg

Image:Ma. State Lottery Ticket 1758.jpg

Source: http://en.wikipedia.org/w/ind http://en.wikipedia.org/w/index.php?title=File:Ma._State_Lottery_Ticket_ ex.php?title=File:Ma._State_Lottery_Ticket_1758.jpg 1758.jpg  License: GNU Free Documentation License

Contributors: User:Ronshelley Image:1776 Continental Congress Lottery Ticket 001.jpg

Source: http://en.wikipedia.org/w/index http://en.wikipedia.org/w/index.php?title=File:1776_Continental_ .php?title=File:1776_Continental_Congress_Lottery_Ticket_001.jpg Congress_Lottery_Ticket_001.jpg  License: GNU Free

Documentation License Contributors: User:Ronshelley

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Image:Harvard Lottery 1811.jpg Image:Louisiana Lottery.jpg

Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?title=File:Louisiana_Lott dex.php?title=File:Louisiana_Lottery.jpg ery.jpg  License: GNU Free Documentation License Contributors: User:Ronshelley

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File:LottoFIN.jpg

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Image:Lottery patent header.JPG

en.wikipedia

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File:Flag of Spain.svg

Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?title=File:Flag_of_the_United_ dex.php?title=File:Flag_of_the_United_Kingdom.svg Kingdom.svg  License: Public Domain Contributors: User:Zscout370 Source: http://en.wikipedia.org/w/ind http://en.wikipedia.org/w/index.php?title=File:Flag_of_Germany.svg ex.php?title=File:Flag_of_Germany.svg  License: Public Domain Contributors: User:Pumbaa80

File:Flag of the United Kingdom.svg File:Flag of Germany.svg

File:Flag of the Netherlands.svg File:Flag of Ireland.svg

Source: http://en.wikipedia.org/w/ind http://en.wikipedia.org/w/index.php?title=File:Flag_of_the_Netherlands.svg ex.php?title=File:Flag_of_the_Netherlands.svg  License: Public Domain Contributors: User:Zscout370

Source: http://en.wikipedia.org/w/in http://en.wikipedia.org/w/index.php?title=File:Flag_of_Ireland.svg dex.php?title=File:Flag_of_Ireland.svg  License: Public Domain Contributors: User:SKopp

Source: http://en.wikipedia.org/w/ind http://en.wikipedia.org/w/index.php?title=File:Flag_of_the_Philippi ex.php?title=File:Flag_of_the_Philippines.svg nes.svg  License: Public Domain Contributors: Fry1989, Homo lupus, Icqgirl, Kallerna, Klemen Kocjancic, Ludger1961, Mattes, Pumbaa80, Slomox, Srtxg, ThomasPusch, Wikiborg, Zscout370, 24 anonymous edits File:Flag of the Philippines.svg File:Flag of Brazil.svg

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License

License Creative Commons Attribution-Share Alike 3.0 Unported http://creativecommons.org/licenses/by-sa/3.0/ 

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