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6 ( six) is the following 5 and preceding 7.

The for 10006 is (E), and for its reciprocal (a).

In mathematics
6 is the smallest positive integer which is neither a nor a . Six is the second smallest ; its proper are , and .

Since six equals the sum of its proper divisors, six is the smallest , , and $\mathcal\left\{S\right\}$-perfect number.

(2018). 9781848000001, Copernicus.

As a perfect number:

• 6 is related to the 3, since . (The next perfect number is 28.)
• 6 is the only even that is not the sum of successive odd cubes.David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
• 6 is the root of the 6-aliquot tree, and is itself the of only one number; the , .

Six is the only number that is both the sum and the product of three consecutive positive numbers.Peter Higgins, Number Story. London: Copernicus Books (2008): 12

Unrelated to 6 being a perfect number, a of length 6 is a "perfect ruler".Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72 Six is a .

Six is the first discrete biprime (2 × 3) and the first member of the (2 × q) discrete family.

Six is a unitary perfect number, a harmonic divisor number and a superior highly composite number, the last to also be a . The next superior highly composite number is . The next primorial is .

There are no Graeco-Latin squares with order 6. If n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n.

There is not a prime p such that the multiplicative order of 2 modulo p is 6, that is, ord p(2) = 6. By Zsigmondy's theorem, if n is a natural number that is not 1 or 6, then there is a prime p such that ord p(2) = n. See for such p.

The ring of integer of the sixth cyclotomic field , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where $\omega = \frac\left\{1\right\}\left\{2\right\}\left(-1 + i\sqrt 3\right) = e^\left\{2\pi i/3\right\}$.

The smallest non- group is the S3 which has = 6 elements.

S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) , the of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n objects on an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the between the sets. This category has a non-trivial to itself only for .

Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest of the plane is obtained by extending this pattern to the lattice in which each circle touches just six others.

A cube has 6 faces
6 is the largest of the four .

A six-sided is a hexagon, one of the three regular polygons capable of . representing hexagons (including six) are called . Because 6 is the product of a power of 2 (namely 21) with nothing but distinct (specifically 3), a regular hexagon is a constructible polygon.

Six is also an octahedral number. It is a triangular number and so is its square ().

There are six basic trigonometric functions.

There are six convex regular polytopes in four .

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.

All primes above 3 are of the form 6 n ± 1 for n ≥ 1.

List of basic calculations
6 × x6121824303642485460 !667278849096102108114120 !1503006006000

6 ÷ x6321.51.210.0.750.0.6 !0.0.50.0.0.4
x ÷ 60.10.0.50.0.811.11.1.51. !1.822.12.2.5

6636216129677764665627993616796161007769660466176 !362797056217678233613060694016
x164729409615625466561176492621445314411000000 !177156129859844826809

Greek and Latin word parts

Hexa
Hexa is classical for "six". Thus:
• "" combines hexa- with the Latinate decimal to name a of 16
• A is a with six sides
• L'Hexagone is a French nickname for the continental part of Metropolitan France for its resemblance to a regular hexagon
• A is a with six faces, with a cube being a special case
• is a poetic form consisting of six feet per line
• A "hex nut" is a nut with six sides, and a hex has a six-sided head
• The prefix "hexa-" also occurs in the of many chemical compounds, such as which has 6 carbon atoms ().

The prefix sex-
Sex- is a prefix meaning "six". Thus:
• Senary is the ordinal adjective meaning "sixth"
• People with have six fingers on each hand
• The measuring instrument called a got its name because its shape forms one-sixth of a whole
• A group of six musicians is called a
• Six babies delivered in one birth are
• pairs – Prime pairs differing by six are sexy, because sex is the Latin word for six.
(2018). 9781448651702, CreateSpace Independent Publishing Platform. .

Evolution of the glyph
The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the of , who wrote it in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66

On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a "b" is not practical.

Just as in most modern , in typefaces with the 6 character usually has an ascender, as, for example, in .

This numeral resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.

In music

In artists

In instruments
• A standard has six strings
• Most instruments have six basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts

In music theory
• There are six whole tones in an octave.
• There are six semitones in a tritone.

In works
• "Six geese a-laying" were given as a present on the sixth day in the popular , "The Twelve Days of Christmas."
• Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of 's oeuvre.
• The theme of the sixth album by Dream Theater, Six Degrees Of Inner Turbulence, was the number six: the album has six songs, and the sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.
• gave six elements of tragedy, the first of which is .

In religion
• There are 6:
• Points on a Star of David
• Orders of the
• Symbolic foods placed on the Passover Seder Plate
• took six days to create the world in the Book of Genesis; humankind was created on day 6. In the City of God, Augustine of Hippo suggested ( book 11, chapter 30) that God's creation of the world took six days because 6 is a .
• The holiday of starts on the sixth day of the month of
• have six wings.
• In Islam:
• There are Six articles of belief
• Fasting six days of , together with the month of , is equivalent to fasting the whole year
• In Hindu theology, a trasarenu is the combination of six celestial paramānus (atoms).

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• Six Lines of a
• Six Ministries of Huang Di

In science

Astronomy

Biology

Chemistry
• A benzene molecule has a ring of six carbon atoms.
• 6 is the of carbon.
• The sixfold of arises from the crystal structure of .
• A is an oligomer made of six subunits.

Medicine
• There are six tastes in traditional Indian Medicine called : sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.
• Phase 6 is one of six pandemic influenza phases.

Physics
• In the Standard Model of particle physics, there are six types of and six types of .
• In statistical mechanics, the has six possible configurations of arrows at each vertex
• There are six colors in the RGB color wheel: (primary) red, blue, green, (secondary) cyan, magenta, and yellow. (See )
• In three-dimensional , there are six unknown support reactions for a statically determinate structure: one in each of the three dimensions, and one moment through each of three possible planes.

In sports
• The teams in the National Hockey League are Toronto, Chicago, Montreal, New York, , and Detroit. They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967
• Number of players:
• In association football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
• In , the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.
• In , the number of players per team, including the goaltender, that are on the ice at any one time during regulation play, excluding penalty situations. (Some leagues reduce the number of players on the ice during overtime.)
• In , six players from each team on each side play against each other
• is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad
• Also in volleyball, standard rules only allow six total substitutions per team per set. (Substitutions involving the libero, a defensive specialist who can only play in the back row, are not counted against this limit.)
• Scoring:
• In both American and Canadian football, 6 points are awarded for a touchdown.
• In Australian rules football, 6 points are awarded for a goal, scored when a kicked ball passes between the defending team's two inner goalposts without having been touched by another player.
• In most competitions (but not the European , which uses static squad numbering), the jersey number 6 is worn by the starting (Southern Hemisphere term) or (Northern Hemisphere term).
• In , the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)

In technology
• On most phones, the 6 key is associated with the letters M, N, and O, but on the it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
• The "6-meter band" in includes the frequencies from 50 to 54 MHz
• 6 is the resin identification code used in recycling to identify

In calendars

In the arts and entertainment

Games
• The number of sides on a cube, hence the highest number on a standard
• The six-sided tiles on a hex grid are used in many tabletop and .
• The highest number on one end of a standard

Comics and cartoons
• The Super 6, a 1966 animated cartoon series featuring six different super-powered heroes.
• The are the heroes of the eponymous animated series
• is a group of super villains who appear in American comic books published by Marvel Comics

Literature
• The Power of Six is a book written by Pittacus Lore, and the second in the series.
• Number 6 is a character in the book series

TV
• Number Six (), is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica
• Number 6, the main protagonist in played by , and portrayed by in the remake.
• Six is a character in the television series Blossom played by Jenna von Oÿ.
• Six is the nickname of Kal Varrik, a central character in the television series Dark Matter, played by .
• Six is a History channel series that chronicles the operations and daily lives of SEAL Team Six.

Movies
• Number 6 () is a character in the movie I Am Number Four (2011)

Anthropology
• The name of the smallest group of and Guiding's equivalent Brownies, traditionally consisting of six people and is led by a "sixer".
• A is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead
• There are said to be no more than six degrees of separation between any two people on Earth.
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac
• Six human physical needs: , , , , , and
• The form part of Chinese history
• 6 is a lucky number in Chinese culture.
• The were a British miscarriage of justice, held in prison for 16 years.
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.

In other fields
• Six pack is a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.
• The fundamental flight instruments lumped together on a cockpit display are often called the Basic Six or six-pack.
• The number of dots in a cell.
• Extrasensory perception is sometimes called the "sixth sense".
• is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.
• In the U.S. Army "Six" as part of a radio call sign is used by the commanding officer of a unit, while subordinate platoon leaders usually go by "One".
(1983). 9780552124195, Corgi Books.

• The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66–68
• A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69

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