2006 PBA Philippine Cup Playoffs

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

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 , and .

Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers. It is also the smallest Granville number, or $\mathcal$-perfect number.

As a perfect number:
*6 is related to the Mersenne prime 3, since . (The next perfect number is 28.)
*6 is the only even perfect number that is not the sum of successive odd cubes.
*6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, .

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

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

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

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 $\omega = \frac(-1 + i\sqrt 3) = e^$.

The smallest non-abelian group is the symmetric group ''S''₃ which has 3! = 6 elements.

''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 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. 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.

6 is the largest of the four all-Harshad numbers.

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.

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 dimensions.

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.

6 is a pronic number and the only semiprime to be.

There are six different 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.

List of basic calculations

Greek and Latin word parts

'

' is classical Greek for "six". Thus:
*"Hexadecimal" combines ' with the Latinate ' to name a number base of 16
*A hexagon is a regular polygon with six sides
**' is a French nickname for the continental part of Metropolitan France for its resemblance to a regular hexagon
*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 "" also occurs in the systematic name of many chemical compounds, such as hexane which has 6 carbon atoms ().

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 Arabic digit

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. 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 artists

*' ("The Six" in English) was a group consisting of the French composers , , , , and in the 1920s
* Bands with the number six in their name include Six Organs of Admittance, 6 O'Clock Saints, Electric Six, Eve 6, Los Xey (' is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer, Slant 6, Vanity 6, and You Me At Six
* #6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot

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 works

*"Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas."
*Divided in six arias, ''Hexachordum Apollinis'' is generally regarded as one of the pinnacles of Johann Pachelbel'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.
*Aristotle gave six elements of tragedy, the first of which is Mythos.

In religion

*In Judaism:
**Six points on a Star of David
**Six orders of the Mishnah
**Six symbolic foods placed on the Passover Seder Plate
**God took six days to create the world in the Old Testament 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 perfect number.
**The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan
**Seraphs have six wings.
*In Islam:
**There are Six articles of belief
**Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year
*In Hindu theology, a '' trasarenu'' is the combination of six celestial (atoms).
*In Taoism:
**Six Lines of a Hexagram
**Six Ministries of Huang Di

In science

Astronomy

*Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster
*The New General Cataloguebr>object
NGC 6, a spiral galaxy Andromeda
*The Roman numeral VI:
**Stands for subdwarfs in the Yerkes spectral classification scheme
**(Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)
*6 Hebe

Biology

*The cells of a beehive are six-sided.
*Insects have six legs.
*Six kingdoms in the taxonomic rank below domain (biology); Animalia, Plantae, Fungi, Protista, Archaea/Archaeabacteria, and Bacteria/Eubacteria. See Kingdom (biology).
*The six elements most common in biomolecules are called the CHNOPS elements; the letters stand for the chemical abbreviations of carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur. See CHON.

Chemistry

*A benzene molecule has a ring of six carbon atoms.
*6 is the atomic number of carbon.
*The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice.
*A hexamer is an oligomer made of six subunits.

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 statistical mechanics, the six-vertex model 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 Tertiary color)
*In three-dimensional Euclidean space, there are six unknown support reactions for a statically determinate structure: one force in each of the three dimensions, and one moment through each of three possible orthogonal planes.

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.)
** <