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Luck 2

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.

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Luck 3

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 thesereligions 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, moresatisfied 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 4

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

A National Lottery "play here!" sign outside a

newsagents on the Euston Road, London.

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, iscloser to an art than to a science, yet numerologists, astrologists or

psychics may disagree. It is interrelated to astrology, and to some

degree to parapsychology and spirituality and is based on

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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Luck 5

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, Daikokutenand 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, by any way, get grapes by the seed of acacia.

2. Luck is but lack of self confidence and fruit of idleness.

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Luck 6

See also

• Chance

• Fate

• Probability

• Serendipity

• Lucky Symbols

External links

• Luck, Destiny, Fate, Karma, or Self-Made? [21] with psychologist Richard Wiseman

• "Lucky": Documentary with Richard Wiseman [22] transcript with link to 10 minute video.

References

[1] http://wordnetweb. princeton.  edu/perl/webwn?s=luck "an unknown and unpredictable phenomenon that causes an event to result one way

rather than another"

[2] http://dictionary. 

reference. 

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://en.  wiktionary.  org/wiki/luck

[4] The 3000 Most Commonly Used Words in the United States (http://www. paulnoll.  com/Books/Clear-English/words-23-24-hundred.  html)

Luck is listed at 2361.

[5] Wikiquote: Luck

[6] http://www. buzzle.  com/articles/twenty-good-luck-quotes.  html

[7] http://www. thaliatook.  com/OGOD/fortuna.  html

[8] Elbow Room (http://books. 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 News 4.15 (http://www.dartmouth. edu/~chance/chance_news/recent_news/chance_news_4.  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 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 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.  manchester. ac. uk/staff/alex.

wood/Luck. 

pdf) Personality and Individual Differences, 45 , 655-660.[15] (http://www. buddhanet.  net/e-learning/qanda09.  htm)

[16] (http://www. thailandlife. com/amulet.  html)

[17] http://ebible. org/web/Isaiah.htm#C65V11

[18] http://www. islamweb. net/ver2/Fatwa/ShowFatwa.  php?lang=E&Id=91036&Option=FatwaId

[19] God creates by wisdom and love #295 (http://www. vatican.  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 FOR PERSONS AND THEIR GOODS #2413 (http://www. 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] http://www. fastcompany.  com/magazine/72/realitycheck.html

[22] http://www. abc. net. au/catalyst/stories/2220191.htm

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Lottery 7

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.

National Lottery building located on Paseo de la

Reforma in Mexico City.

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" draw

where the organizers promise that the prize will be 50% of the revenue.

Many recent lotteries allow purchasers to select the numbers on thelottery 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.

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 totalprize 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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Lottery 8

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 goodworkes." 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]

English Lottery 1566 Scroll.

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. Mostpeople 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 State Lottery Ticket 1814 issued by broker Swift & Co.

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, theEnglish lotteries ran for over 250 years, until the

government, under constant pressure from the

opposition in parliament, declared a final lottery in

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

Ticket from an 1814 lottery to raise money for Queen's College, New

Jersey.

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

in the financing of both private and public ventures. It

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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Lottery 9

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."

Massachusetts Lottery Ticket 1758 French & Indian Wars

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 MountainRoad Lottery in 1768 was unsuccessful. However,

these rare lottery tickets bearing George Washington's

signature have become collectors' items which sold for

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.

1776 Lottery ticket issued by Continental Congress to finance

Revolutionary War.

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.

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 May1795), and Harvard College (lottery in March 1806). Many American churches raised building funds through state

authorized private lotteries.

Harvard Lottery Ticket 1811

However, lotteries eventually became a cause of 

financial mismanagement and scandal. Most notorious

was the Louisiana State Lottery (1868 – 1892) which

was aptly called the "Golden Octopus" because its

tentacles reached into every home in America.

Louisiana Lottery 1/20th of a $20 ticket: The Last of the Lotteries

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

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 broughta complete halt to all lotteries in the United States by 1900.

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Lottery 10

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.

New Hampshire Lottery Ticket 1964

For modern USA lotteries visit: Lotteries in the United States

Countries with a national lottery

This maneki neko beckons customers to purchase takarakuji tickets

in Tokyo, Japan.

Africa

• South Africa: South African National Lottery• Kenya: Toto 6/49, Kenya Charity Sweepstakes

North and South America

• Argentina: Quiniela, Loto and various others

• Barbados: Barbados lottery and various others

• Brazil: Mega-Sena and various others

• Canada: Lotto 6/49 and Lotto Max

• Chile: Polla Chilena de Beneficencia S.A.

• Costa Rica: Lotería Nacional, Chances LoteríaPopular, Lotería Tiempos, and Lotería Instantanea

(better known as "Raspaditas" since the tickets are

scratch cards).

• El Salvador: Lotería Nacional de Beneficencia,

Lotín (scratch cards).

• Dominican Republic: Lotería Electrónica

Internacional Dominicana S.A.

• Ecuador: Lotería Nacional

• Mexico: Lotería Nacional para la Asistencia Pública

and Pronósticos para la Asistencia Pública

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Lottery 11

Asia

• Mainland China: China Welfare Lottery, China Sports Lottery

• Hong Kong: Mark Six

• Macau: Macau SLOT

• Taiwan: Taiwan Lottery

• Israel: Lotto• Japan: Takarakuji

• Lebanon: La Libanaise des Jeux

• Malaysia: Sports Toto Malaysia, Magnum Corporation, Pan Malaysian Pools)

• Philippines: Philippine Lotto Draw

• Singapore: Singapore Pools

• South Korea: Lotto

• Sri Lanka: National Lottery, Development Lottery

• Thailand: สลากกินแบง่รั ฐบาล (salak gin bang ratthabarn or "Government Lottery"), also called lottery or หวย(huay).

• Vietnam: Xổ số kiến thiết

Australasia

• New Zealand: NZ Lotteries

• Australia: OZ Lotto, Powerball, Lotto

Europe

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.}

• Pan-European: EuroMillions

• Nordic countries: Viking Lotto

• Austria: Lotto 6 aus 45, EuroMillions and Zahlenlotto• Belgium: Loterie Nationale or Nationale Loterij and EuroMillions

• Bulgaria: Durzhavna lotariya, TOTO 2 (6/49, 6/42, 5/35)

• Croatia: Hrvatska lutrija

• Czech Republic: Sazka

• Denmark: Lotto, Klasselotteriet

• Finland: Lotto, scratch tickets, racing & football pools (Veikkaus)

• France: La Française des Jeux and EuroMillions

• Germany: Lotto 6 aus 49, Spiel 77 and Super 6

• Greece: OPAP (Greek: ΟΠΑΠ – Οργανισμός Προγνωστικών

 Αγώνων Ποδοσφαίρου), Lotto 6/49, Joker 5/45 + 1/20 and variousothers

• Hungary: Lottó

• Iceland: Lottó

• Ireland: The National Lottery (Irish: An Chrannchur Náisiúnta) and

EuroMillions

• Italy: Lotto and SuperEnalotto

• Latvia: Latloto 5/35, SuperBingo, Keno

• Liechtenstein: International Lottery in Liechtenstein Foundation

• Luxembourg: EuroMillions

• Malta: Super 5 and Lotto

• Macedonia: Lotarija na Makedonija

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Lottery 12

• Montenegro: Lutrija Crne Gore

• Netherlands: Nationale Postcode Loterij, Staatsloterij (The State Lottery)

• Norway: Lotto

• Poland: Lotto

• Portugal: Lotaria Clássica, EuroMillions and Lotaria Popular

• Romania: Loteria Română (6/49, 5/40, Joker)

• Russia: Gosloto (Russian: Гослото, The State Lottery)

• Serb Republic: Lutrija Republike Srpske

• Serbia: Državna Lutrija Srbije (The State Lottery of Serbia)

• Slovakia: Loto

• Slovenia: Loterija Slovenije

• Spain: Loterías y Apuestas del Estado, EuroMillions and ONCE

• Catalonia: Loteria de Catalunya (6/49 amongst others)

• Sweden: Lotto (Svenska Spel)

• Switzerland: Swiss Lotto and EuroMillions

• Turkey: Various games by Milli Piyango İdaresi (National Lottery Administration) including Loto 6/49 and jackpots

• Ukraine: Super Lotto

• United Kingdom: The National Lottery, the main game being Lotto. Also 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.

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Lottery 13

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 appealthis 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 Lottery Corporation (New Brunswick, Nova Scotia, Prince Edward Island, Newfoundland and Labrador)

• Loto-Québec (Quebec)

• Ontario Lottery and Gaming Corporation (Ontario)

• Western Canada Lottery Corporation (Manitoba, Saskatchewan, Alberta, Yukon Territory, Northwest Territories,Nunavut)

• British Columbia Lottery Corporation (British 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 one of the most 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 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 14

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 thelotteries. 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 theInternet 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 16

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 numbers are 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 "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, which 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 anextreme case, giving a very small payout (US$2) even if a player matches only the final Mega Ball 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 lotteryticket is a poor economic decision. Indeed, after taking into account the present value of a given lottery prize as a

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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 UnitedStates

Won by one ticket holder fromNew Jersey and one fromGeorgia

6 March2007

World's largest jackpot

$365m Powerball UnitedStates

One ticket bought jointly byeight co-workers at a Nebraskameat processing plant

18 February2006

World's largestsingle ticket winner

$363m The Big Game UnitedStates

Two winning tickets: Larry andNancy Ross (Michigan), Joeand Sue Kainz (Illinois)

9 May 2000 The Big Game isnow named MegaMillions

$314.9 Million Powerball UnitedStates

Andrew Jackson “Jack”Whittaker, Jr.

2002-12-25 World's largestsingle person winner

€180m EuroMillions France ×2,  Portugal ×1

Three ticket holders 3 February2006

Europe's largest jackpot

€147,8m SuperEnalotto Italy One ticket holder from Bagnone(Toscana)

22 August2009

Europe's largestsingle winner

€126m EuroMillions Spain Anonymous 25 year-old woman 8 May 2009 Largest singlewinner inEuroMillions.

£42m National Lottery UnitedKingdom

Three ticket holders 6 January1996

€37.6m National Lottery Germany Won by a nurse from NorthRhine-Westphalia

7 October2006

Largest Germanprize and singlewinner

€25m State Lottery Netherlands Ticket sold in The Hague 10 July2008

Tax free lump sum

€115m Euro Millions Ireland Dolores McNamara August 2005 Biggest singlewinner and jackpot(Ireland) Tax freelump sum

€42m Jordan Banks August 2008 Tax free lump sum

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Lottery 18

₱347.8m(US$4.765m)

Philippine LottoDraw

  Philippines Two winners in Luzon 22 February2009

Asia's largest prize

R$145m Mega-Sena   Brazil Won by one ticket holder fromBrasília (Federal District) andone from Santa Rita do PassaQuatro (São Paulo)

31December2009

South America'slargest prize

£56m EuroMillions UnitedKingdom

Nigel Page and his partnerJustine Laycock. The total jackpot of £113m was sharedwith a winner in Spain.

12 January2010

Britain's biggest everlottery prize

Sources:

http://www.  usamega. com/archive-052000. htm

http://www.  timesonline. co. uk/tol/news/world/europe/article6274441.  ece

http://news.  bbc. co. uk/1/hi/world/europe/4746057.stm

http://news.  bbc. co. uk/1/hi/uk/4676172.  stm

http://news.  bbc. co. uk/1/hi/world/americas/4740982.  stm

http://www.  sisal.  it/se/se_main/1,4136,se_Record_Default,00.  html

http://www.  howtomakeabilliondollars. com/145-million-european-lottery-this-weekend/ 

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 thefinal 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.

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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 feelanomie 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 pool

• Combinatorial number system• Gaming mathematics

• GTech Corporation

• Intralot

• Keno

• Lottery Wheeling

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Lottery 20

Further reading

• A History of English Lotteries, by John Ashton, London: Leadenhall Press, 1893

• Fortune's Merry Wheel, by John Samuel Ezell, Harvard University Press, 1960.

• Lotteries and Sweepstakes, 1932 by Ewen L'Estrange

• The Lottery Encyclopedia, 1986 by Ron Shelley (NY Public Library)

• Fate's Bookie: How The Lottery Shaped The World by Gary Hicks, History Press, 2009

External links

• World Lottery Association [15]

• Euler's Analysis of the Genoese Lottery [16]

References

[1] Ron Shelley, The Lottery Encyclopedia(1986)

[2] John Ashton, A History of English Lotteries, 1893.

[3] John Samuel Ezell, Fortune's Merry Wheel, 1960.[4] Two Winners to Share Record A$106 Million Australian Lottery (http://www.bloomberg. com/apps/news?pid=20601081&

sid=ahKl7APqrtD0)

[5] Internet Archive (http://web. archive.  org/web/20060423062311/http://lotteries.  olgc.  ca/consumer_fq.   jsp#qa17)

[6] Prize Divisions (http://www.mylotto. co. nz/wps/wcm/myconnect/lotteries2/nzlotteries/Primary/Our_Games/Lotto/AllAboutLotto/ 

LottoPrizeDivisions.  jsp)

[7] New Zealand State Lotteries Commission (http://www. mylotto.  co. nz/wps/wcm/myconnect/lotteries2/nzlotteries/Global/ 

AboutNZLotteries/StatutoryFunction/)

[8] New Zealand Lotto Results (http://lotto. nzpages. net. nz)

[9] Internet Archive (http://web. archive.  org/web/19981212015337/http://lotto.nzpages.  net. nz/)

[10] Bellhouse, D.R., “The Genoese Lottery”, Statistical Science, vol. 6, No. 2. (May, 1991), pp. 141 -148

[11] Megamillions game history (http://www. megamillions. com/aboutus/game_history.  asp)

[12] http://lottery. merseyworld. com/Wheel/Wheel.html

[13] http://www. dailyrecord.  co. uk/news/tm_objectid=16164112&method=full&siteid=66633&

headline=primary-pupil-s--pound-27m-lotto-win--name_page.  html

[14] "Legalized Gambling; America's Bad Bet" by John Eidsmoe

[15] http://www. world-lotteries. org/ 

[16] http://mathdl. maa. org/convergence/1/?pa=content&sa=viewDocument&nodeId=217&bodyId=93

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Combinatorial number system 21

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, 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 isrepresenting a k -combination by  N  rather than the other way around. Distinct numbers correspond to distinct

k -combinations, and produce them in lexicographic order; moreover the numbers less than correspond to all

k -combinations of { 0, 1, ..., n  − 1}. The correspondence does not depend on the size n of the set that the

k -combinations are taken from, so it can be interpreted as a map from N to the k -combinations taken from N; in this

view the correspondence is a bijection.

The number N corresponding to (ck ,...,c

2,c

1) is given by

The fact that a unique sequence so corresponds to any number  N was observed by D.H. Lehmer.[1]

Indeed a greedyalgorithm finds the k -combination corresponding to N : take c

k maximal with , then take c

k  − 1maximal

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" ciis 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

that is at a given position in the lexicographic ordering, without having to explicitly list the k -combinations

preceding it; this allows for instance random generation of  k -combinations of a given set. Enumeration of k -combinations has many applications, among which software testing, sampling, quality control, and the analysis of 

lottery games.

Ordering combinations

A k -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 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 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 of the set N of all natural numbers, without

explicitly mentioning n.

In order to ensure that the numbers representing the k -combinations of {0, 1, ..., n − 1} are less than those

representing k -combinations not contained in {0, 1, ..., n − 1}, the k -combinations must be ordered in such 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

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Combinatorial number system 22

(this associates distinct numbers to all finite sets of natural numbers); then comparison of  k -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 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.

Place of a combination in the ordering

The number associated in the combinatorial number system of degree k  to a k -combination C  is the number of 

k -combinations strictly less than C in the given ordering. This number can be computed from C = { ck , ..., c

2, c

1}

with ck 

> ... > c2

> c1

as follows. From the definition of the ordering it follows that for each k -combination S strictly

less than C , there is a unique index i such that ci

is absent from S , while ck , ..., c

i+1are present in S , and no other

value larger than ciis. One can therefore group those k -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 , ..., c

i+1in 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 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 strictly less than C then is

and this is the index (starting from 0) of C in the ordered list of k -combinations. Obviously there is for every N  ∈ N

exactly one k -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 immediately. The

reverse process of finding the k -combination at a given place N requires somewhat more work, but is straightforward

nonetheless. By the definition of the lexicographic ordering, two k -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 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

of  N and k .

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 ciwill be the maximal ones with , namely c

i= i − 1.

Thus we have found the 5-combination {8, 6, 3, 1, 0}.

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Combinatorial number system 23

Applications

One could use the combinatorial number system to list or traverse all k -combinations of a given finite set, but this is

a very inefficient way to do that. Indeed, given some k -combination it is much easier to find the next combination in

lexicographic ordering directly than to convert a number to a k -combination by the method indicated above. To find

the next combination, find the smallest i ≥ 2 for which ci ≥ c

i− 1+2; then increase c

i−1by one and set all c

 jwith j < i −

1 to their minimal value j − 1. If the k -combination is represented as a binary value with k bits 1, then the next suchvalue 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(unsigned 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=x'10^a00^b

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

  return false; // signal that next k-combination cannot be

represented

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

x ← x'100^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 at index N has useful applications. Notably,

it allows generating a random k -combination of an n-element set using a random integer  N with ,

simply by converting that number to the corresponding k -combination. If a computer program needs to maintain a

table with information about every k -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 number system (also called factoradics)

References

[1]  Applied Combinatorial Mathematics , Ed. E. F. Beckenbach (1964), pp.27−30.

[2] Knuth, D. E. (2005), "Generating All Combinations and Partitions", The Art of Computer Programming, 4, Fascicle 3, Addison-Wesley,

pp. 5−6, ISBN 0-201-85394-9.

[3] Pascal, Ernesto (1887), Giornale di Matematiche, 25, pp. 45−49

[4] McCaffrey, James (2004), Generating the mth Lexicographical Element of a Mathematical Combination (http://msdn.  microsoft.  com/ 

en-us/library/aa289166(VS.  71).  aspx), Microsoft Developer Network,

[5] Knuth, D. E. (2009), "Bitwise tricks and techniques", The Art of Computer Programming, 4, Fascicle 1, Addison-Wesley, pp. 54,

ISBN 0-321-58050-8

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Gaming mathematics 24

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 the dice in craps is an experiment that generates events such as occurrences of certain 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 isrepresented by the following set in the experiment of rolling one die: {2, 4, 6}.

• Spinning the roulette wheel is an experiment whose generated events could be the occurrence of a 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.

• Dealing cards 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, the experiment of drawing six numbers from the 49 generate events such as drawing six 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, the experiment of dealing the initial five card hands generates events such as 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 two cards to a player who has discarded two cards is another experiment whose sample 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

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Gaming mathematics 25

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 manycategories 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 related to your own play or to opponents’ play;

• Events related to one person’s play or to several persons’ play;

• Immediate events or long-shot events.

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 amathematical 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 {3, 5} (whose literal definition is occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};

• Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;

• Events {3, 5} and {4} are incompatible or exclusive because their intersection is empty; that is, they cannot occur

simultaneously;

• Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;

• In the experiment of rolling two dice one after another, the 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 of dealing (3♣, 3♦) to a player is an elementary event;

• The event of dealing 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 heartshigher 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:

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Gaming mathematics 26

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 usethe 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 merely pure applications of probability calculus and gaming situations are not just isolatedevents 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.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 theplayer 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 onlypay 4 times the amount wagered for a winning wager.

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Gaming mathematics 27

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 houseedge 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 theshoe (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) = 2b 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 thatyou 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.

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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

ĀryabhaṭaCyrillic

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]

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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 7 6 5 4 3 2 1

place value: 7! 6! 5! 4! 3! 2! 1! 0!

in decimal: 5040 720 120 24 6 2 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 364

51

40

31

20

1, 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, untilthis 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 values1 ⁄ 

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:

decimal factorial

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!

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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! = 72010

, 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 infactoradic form. This mapping has been termed the Lehmer code (or inversion table). For example, with n = 3, such

a mapping is

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Factorial number system 32

decimal factorial permutation

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 2982

10) pick out the digits in (4,0,6,2,1,3,5), the 2982nd permutation of the numbers 0 through 6.

4041000! → (4,0,6,2,1,3,5)

factoradic: 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 factoradics permutation pair

010

000!000

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

110

000!010

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

...

510

000!210

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

610

010!000

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

710

010!010

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

...

2210

110!200

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

...

3410

210!200

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

3510

210!210

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

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Factorial number system 33

External links

• Mantaci, Roberto; Rakotondrajao, Fanja (2001), "A permutation representation that knows what “Eulerian”

means" [7] (PDF), Discrete Mathematics and Theoretical Computer Science 4: 101 – 108.

• Arndt, Jörg (March 5, 2009). Algorithms for Programmers: Ideas and source code (draft) [8]. pp. 224 – 234.

See also

• Combinatorial number system (also called combinadics)

• Factorial

External links

• A Lehmer code calculator [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. E. (1973), "Volume 3: Sorting and Searching", The Art of Computer Programming, Addison-Wesley, pp. 12, ISBN 0-201-89685-0

[2] Cantor, G. (1869), Zeitschrift für Mathematik und Physik , 14.

[3] Knuth, D. E. (1997), "Volume 2: Seminumerical Algorithms", The Art of Computer Programming (3rd ed.), Addison-Wesley, pp. 192,

ISBN 0-201-89684-2.

[4] Laisant, Charles-Ange (1888), "Sur la numération factorielle, application aux permutations" (http://www. numdam.  org/ 

item?id=BSMF_1888__16__176_0) (in French), Bulletin de la Société Mathématique de France 16: 176 – 183, .

[5] The term "factoradic" is apparently introduced in McCaffrey, James (2003), Using Permutations in .NET for Improved Systems Security

(http://msdn2.  microsoft. 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://en.  wikipedia.  org/wiki/Oeis%3Aa007623

[7] http://www. dmtcs. org/volumes/abstracts/pdfpapers/dm040203.  pdf 

[8] http://www.  jjj. de/fxt/#fxtbook

[9] http://www-ang. kfunigraz.  ac.at/~fripert/fga/k1lehm.  html

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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

ĀryabhaṭaCyrillic

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 a useful set of numbers (e.g. all integers, or rational numbers)• Give every number represented a unique representation (or at least a standard representation)

• Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual 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.

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Numeral system 35

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 importancehere, 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. 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 notation, not to be confused with p-adic numbers. Bijective base-1

is the same as unary.

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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 − 1

bn − 1 + an − 2

bn − 2 + ... + a0b0 and writing the enumerated digits a

na

n − 1a

n − 2... a

0in 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: numberbase

. 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 1×21

+ 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 digitsrequired 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 1 0.1 0.01 ...

Decimal example digit 4 3 2 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 thebase. 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.

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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,

representing 0 – 25 and 26 – 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), 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

• Babylonian numerals – a sexagesimal (base-60) system

• Computer numbering formats

• Golden ratio base

• List of numeral system topics

• Maya numerals – a base-20 system• N-ary

• Number names

• Quipu

• Recurring decimal

• Residue number system

• Subtractive notation

References

[1]  Hindu Arabic Numerals by David Eugene Smith Google Books) (http://books.  google.  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)

• Georges Ifrah. 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, "Positional

Number Systems".

• A. L. Kroeber (Alfred Louis Kroeber) (1876 - 1960), Handbook of the Indians of California, Bulletin 78 of the

Bureau of American Ethnology of the Smithsonian Institution (1919)

• J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and

Chicago, 1997.

• Hans J. Nissen, P. Damerow, R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993, ISBN0-226-58659-6.

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Numeral system 38

• Denise Schmandt-Besserat, How Writing Came About , University of Texas Press, 1992, ISBN 0-292-77704-3.

• Claudia Zaslavsky, Africa Counts: Number and Pattern in African Cultures , Lawrence Hill Books, 1999, ISBN

1-55652-350-5.

External links

• Numerical Mechanisms and Children's Concept of Numbers (http://web. media. mit.edu/~stefanm/society/ som_final. html)

• Software for converting from one numeral system to another (http://billposer. org/Software/libuninum.  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 point, mixed radix, base (mathematics)

• Unary numeral system (base 1)

• Tally marks

• Binary numeral system (base 2)

• Negative base numeral system (base −2)

• Ternary numeral system numeral system (base 3)

• Balanced ternary numeral system (base 3)

• Negative base numeral system (base −3)

• Quaternary numeral system (base 4)• Quater-imaginary base (base 2√−1)

• Quinary numeral system (base 5)

• Pentimal system

• Senary numeral system (base 6)

• Septenary numeral system (base 7)

• Octal numeral system (base 8)

• Nonary (novenary) numeral system (base 9)

• Decimal (denary) numeral system (base 10)

• Bi-quinary coded decimal

• Negative base numeral system (base −10)

• Duodecimal (dozenal) numeral system (base 12)

• Hexadecimal numeral system (base 16)

• Vigesimal numeral system (base 20)

• Sexagesimal numeral system (base 60)

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Article Sources and Contributors 40

Article Sources and ContributorsLuck  Source: http://en.wikipedia.org/w/index.php?oldid=366119802 Contributors: A D Monroe III, Aca d Ronin, Aces&8s, Adashiel, Administrationers, Admoose, Afil, AgentPeppermint,

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,

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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,

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Tealwisp, Ter308, Texture, The Iconoclast, The Wookieepedian, TheSuave, TheTrojanHought, Themagicalpixieelf, Themfromspace, Themissinglint, Tidus the BlitzStar, Tiptoety, Tom harrison,

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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

Lottery  Source: http://en.wikipedia.org/w/index.php?oldid=366804592 Contributors: 159753, 2005, 9Nak, A bit iffy, A little insignificant, Accurizer, Adam Conover, Aervanath, Ahda,

Ahoerstemeier, Aiman abmajid, Alan Liefting, Ale jrb, Andrevan, Andrewpmk, Andrzej P. Wozniak, AnonUser, Antandrus, Anwar saadat, Arenarax, Avala, AxG, B.S. Lawrence, BD2412, BKfi,

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Combinatorial number system  Source: http://en.wikipedia.org/w/index.php?oldid=358361663 Contributors: Eraoul, Giftlite, JamesDmccaffrey, Jwmcleod, Lantonov, Lasloo, LeeHunter, Marc

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Image Sources, Licenses and Contributors 41

Image Sources, Licenses and ContributorsImage:Four-leaf clover.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Four-leaf_clover.jpg  License: GNU Free Documentation License Contributors: User:Phyzome

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Magazine" was published by "the King Publishing Co., Wilmington, Del.".

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Opponent

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Jersey, USA

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License 42

License

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