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, ordp(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 ordp(2) = n. See A112927 for such p.
The ring of integer of the sixth cyclotomic field Q(ζ6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where .
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 nisomorphic 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.
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 hexagonallattice in which each circle touches just six others.
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 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.
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.
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.
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.
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.)
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 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.)
On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry 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
Six Flags 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". (For a similar example see also: Rainbow Six.)
^Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN978-1-84800-000-1.
^"Granville number". OeisWiki. The Online Encyclopedia of Integer Sequences. Archived from the original on 29 March 2011. Retrieved 27 March 2011.