6 (number)
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6 Number

6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.[1]

## In mathematics

Six is the smallest positive integer which is neither a square number nor a prime number; it is the second smallest composite number, behind 4; its proper divisors are 1, 2 and 3.[1]

Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers.[1] It is also the smallest Granville number, or ${\displaystyle {\mathcal {S}}}$-perfect number.[2][3]

As a perfect number:

Six is the only number that is both the sum and the product of three consecutive positive numbers.[5]

Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".[6] Six is a congruent number.[7]

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

Six is a unitary perfect number,[8] a primary pseudoperfect number,[9] a harmonic divisor number[10] and a superior highly composite number,[11] the last to also be a primorial.

There are no Graeco-Latin squares with order 6.[12] 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 ${\displaystyle p}$ such that the multiplicative order of 2 modulo ${\displaystyle p}$ is 6, that is, ${\displaystyle ord_{p}(2)=6}$ By Zsigmondy's theorem, if ${\displaystyle n}$ is a natural number that is not 1 or 6, then there is a prime ${\displaystyle p}$ such that ${\displaystyle ord_{p}(2)=n}$. See A112927 for such ${\displaystyle p}$.

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

The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.[1]

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) Steiner system, the projective plane 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 isomorphic 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 bijections between the sets. This category has a non-trivial functor 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.[13] The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

A cube has six faces.

6 is the largest of the four all-Harshad numbers.[14]

A six-sided polygon is a hexagon,[1] one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon.

Six is also an octahedral number.[15] It is a triangular number[16] and so is its square (36).

There are six basic trigonometric functions.[17]

There are six convex regular polytopes in four dimensions.

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.[18]

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

6 is a pronic number.[19]

### List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 50 100 1000
6 × x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 ÷ x 6 3 2 1.5 1.2 1 0.857142 0.75 0.6 0.6 0.54 0.5 0.461538 0.428571 0.4
x ÷ 6 0.16 0.3 0.5 0.6 0.83 1 1.16 1.3 1.5 1.6 1.83 2 2.16 2.3 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x6 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809

## Greek and Latin word parts

### Hexa

Hexa is classical Greek for "six".[1] Thus:

### The prefix sex-

Sex- is a Latin prefix meaning "six".[1] Thus:

• Senary is the ordinal adjective meaning "sixth"[23]
• People with sexdactyly have six fingers on each hand
• The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole circle
• A group of six musicians is called a sextet
• Six babies delivered in one birth are sextuplets
• Sexy prime pairs - Prime pairs differing by six are sexy, because sex is the Latin word for six.[24][25]

The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).

## Evolution of the Arabic digit

The first appearance of 6 is in the Edicts of Ashoka circa 250 BCE. These are Brahmi numerals, ancestors of Hindu-Arabic numerals.
The first known digit "6" in the number "256" in Ashoka's Minor Rock Edict No.1 in Sasaram, circa 250 BCE

The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to the Brahmi numerals of India, which are first known from the Edicts of Ashoka circa 250 BCE.[26][27][28][29] It was written 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.[30]

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 typefaces, in typefaces with text figures the character for the digit 6 usually has an ascender, as, for example, in .[31]

This digit 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

A standard guitar has six strings.

### In instruments

• A standard guitar has six strings[38]
• Most woodwind 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.[39]
• There are six semitones in a tritone.[40]

## In science

### Biology

The cells of a beehive are six-sided.

### Medicine

• There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.[63]
• Phase 6 is one of six pandemic influenza phases.[64]

### Physics

In the Standard Model of particle physics, there are six types of quarks and six types of leptons.

## In sports

• The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit.[68] 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 box lacrosse, the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.[69]
• In ice hockey, 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.)[70]
• In volleyball:
• Six players from each team on each side play against each other.[71]
• 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.)
• Six-man football 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.[72]
• Scoring:
• In both American and Canadian football, 6 points are awarded for a touchdown.[73]
• 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 cricket, six runs are scored for the batting team when the ball is hit to the boundary or the ground beyond it without having touched the ground in the field.
• In basketball, the ball used for women's full-court competitions is designated "size 6".[74]
• In most rugby league competitions (but not the Super League, which uses static squad numbering), the jersey number 6 is worn by the starting five-eighth (Southern Hemisphere term) or stand-off (Northern Hemisphere term).
• In rugby union, 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.)[75]

## In the arts and entertainment

### Games

• The number of sides on a cube, hence the highest number on a standard die[79]
• The six-sided tiles on a hex grid are used in many tabletop and board games.
• The highest number on one end of a standard domino

### Comics and cartoons

• The Super 6, a 1966 animated cartoon series featuring six different super-powered heroes.[80]
• The Bionic Six are the heroes of the eponymous animated series[81]
• Sinister Six is a group of super villains who appear in American comic books published by Marvel Comics[82]

### Musicals

• Six is a modern retelling of the lives of the six wives of Henry VIII presented as a pop concert.[92]

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55. ^ Rhoads, Samuel E. (1996). The Sky Tonight: A Guided Tour of the Stars Over Hawai'i. Bishop Museum Press. ISBN 978-0-930897-93-2. Three Messier objects are visible in this part of the sky : M6 , M7 and M8 .
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57. ^ Parker, Steve (2005). Ant Lions, Wasps & Other Insects. Capstone. p. 16. ISBN 978-0-7565-1250-7. Insects have six legs each...
58. ^ Pendarvis, Murray P.; Crawley, John L. (2019-02-01). Exploring Biology in the Laboratory: Core Concepts. Morton Publishing Company. p. 10. ISBN 978-1-61731-899-3. ...presently at least six kingdoms are recognized;
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64. ^ Pandemic Influenza Preparedness and Response Guidance for Healthcare Workers and Healthcare Employers. OSHA, U.S. Department of Labor. 2007. p. 8. The WHO Plan describes six phases of increasing public health risk associated with the emergence of a new influenza
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66. ^ Jimbo, M.; Jimbo, Michio; Miwa, Tetsuji; Tsuchiya, Akihiro (1989). Integrable Systems in Quantum Field Theory and Statistical Mechanics. Academic Press. p. 588. ISBN 978-0-12-385342-4. Allowed configurations in the six-vertex model and their statistical weights
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68. ^ Bamford, Tab (2016-10-15). 100 Things Blackhawks Fans Should Know & Do Before They Die. Triumph Books. ISBN 978-1-63319-638-4. the Original Six
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90. ^
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• A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
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