# 6

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

 ← 5 6 7 →
Cardinalsix
Ordinal6th
(sixth)
Numeral systemsenary
Factorization2 × 3
Divisors1, 2, 3, 6
Greek numeralϚ´
Roman numeralVI, vi, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Senary106
Octal68
Duodecimal612
Greekστ (or ΣΤ or ς)
Arabic, Kurdish, Sindhi, Urdu٦
Persian۶
Amharic
Bengali
Chinese numeral六，陸
Devanāgarī
Gujarati
Hebrewו
Khmer
Thai
Telugu
Tamil
Saraiki٦
Malayalam

## 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 after four, equal to the sum and the product of its three proper divisors (1, 2 and 3). As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors. It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).

6 is a pronic number and the only semiprime to be. It is the first discrete biprime (2 × 3) which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.

As a perfect number:

Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two.

All integers $n$  that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or ${\mathcal {S}}$ -perfect number.

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

6 is the second primary pseudoperfect number, and harmonic divisor number. It is also the second superior highly composite number, and the last to also be a primorial.

There are 6 non-equivalent ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).

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 A112927 for such $p$ .

The ring of integer of the sixth cyclotomic field Q6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where $\omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}$ .

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

There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.

The smallest non-abelian group is the symmetric group $\mathrm {S_{3}}$  which has 3! = 6 elements.

Six is a triangular number and so is its square (36). It is the first octahedral number, preceding 19.

A six-sided polygon is a hexagon, 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 with a compass and straightedge alone. A hexagram is a six-pointed geometric star figure (with the Schläfli symbol {6/2}, 2{3}, or {{3}}).

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 sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

There is only one non-trivial magic hexagon: it is of order-3 and made of nineteen cells, with a magic constant of 38. All rows and columns in a 6 × 6 magic square collectively generate a magic sum of 666 (which is doubly triangular). On the other hand, Graeco-Latin squares with order 6 do not exist; if $n$  is a natural number that is not 2 or 6, then there is a Graeco-Latin square of order $n$ .

The cube is one of five Platonic solids, with a total of six squares as faces. It is the only regular polyhedron that can generate a uniform honeycomb on its own, which is also self-dual. The cuboctahedron, which is an Archimedean solid that is one of two quasiregular polyhedra, has eight triangles and six squares as faces. Inside, its vertex arrangement can be interpreted as three hexagons that intersect to form an equatorial hexagonal hemi-face, by-which the cuboctahedron is dissected into triangular cupolas. This solid is also the only polyhedron with radial equilateral symmetry, where its edges and long radii are of equal length; its one of only four polytopes with this property — the others are the hexagon, the tesseract (as the four-dimensional analogue of the cube), and the 24-cell. Only six polygons are faces of non-prismatic uniform polyhedra such as the Platonic solids or the Archimedean solids: the triangle, the square, the pentagon, the hexagon, the octagon, and the decagon. If self-dual images of the tetrahedron are considered distinct, then there are a total of six regular polyhedra that are formed by three different Weyl groups in the third dimension (based on tetrahedral, octahedral and icosahedral symmetries).

How closely the shape of an object resembles that of a perfect sphere is called its sphericity, calculated by:

$\Psi ={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}={\frac {A_{s}}{A_{p}}},$

where $A_{s}$  is the surface area of the sphere, $V_{p}$  the volume of the object, and $A_{p}$  the surface area of the object.

In four dimensions, there are a total of six convex regular polytopes: the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell.

$\mathrm {S_{6}}$ , with 720 = 6! 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, the four-dimensional 5-cell, 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 $n=6$ .

In the classification of finite simple groups, twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice, and eight third generation subgroups of the friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs (Ly, O'N, Ru, J4, J3, and J1).

### 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". Thus:

• "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16
• A hexagon is a regular polygon with six sides
• A hexahedron is a polyhedron with six faces, with a cube being a special case
• Hexameter is a poetic form consisting of six feet per line
• A "hex nut" is a nut with six sides, and a hex bolt has a six-sided head
• The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as hexane which has 6 carbon atoms (C6H14).

### The prefix sex-

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

• Senary is the ordinal adjective meaning "sixth"
• 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.

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

## Evolution of the Hindu-Arabic digit The first appearance of 6 is in the Edicts of Ashoka c. 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, c. 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 c. 250 BCE. 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.

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  .

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

### In instruments

• A standard guitar has six strings
• 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.
• There are six semitones in a tritone.

## In religion

### Islam

Indeed, We created the heavens and the earth and everything in between in six Days,1 and We were not ˹even˺ touched with fatigue.2

— Surah Qaf:38

Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to Surah Al-Hajj (The Pilgrimage):47, a heavenly Day is 1000 years of our time. The Day of Judgment will be 50,000 years of our time - Surah Al-Maarij (The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah.

Note 2: Some Islamic scholars believe this verse comes in response to Exodus 31:17, which says, “The Lord made the heavens and the earth in six days, but on the seventh day He rested and was refreshed.”

## In science

### 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.
• Phase 6 is one of six pandemic influenza phases.

### 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. 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.
• 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.)
• In volleyball:
• Six players from each team on each side play against each other.
• 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.
• 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 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".
• 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.)

## In the arts and entertainment

### Games

• The number of sides on a cube, hence the highest number on a standard die
• 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.

### Musicals

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