# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 6

- 6 is the only even evil perfect number (submitted by Alexey Radul)
- 6 is the order of the smallest non-abelian group (submitted by Sam Steingold)
- 6 is the only number (except 1) such that the sum of all the primes up to 6 equals the sum of all the composite numbers up to 6 (inclusive)
- 6 is the only even perfect number, for which repeatedly summing the digits you do not get 1
- 6 is the only mean between a pair of twin primes which is triangular
- The symmetric group S
_{6}is the only finite symmetric group which has an outer automorphism - 6 is the minimum number of colors that is always sufficient to color any map on a Klein bottle or on a Möbius strip
- 6 is the smallest perfect number
- 6 is the only number that is both the sum and the product of the same three distinct positive integers
- 6 is the largest order for which Graeco-Latin squares do not exist
- 6 is the only square-free perfect number (submitted by Alexey Radul)
- A web page about 6: Some Thoughts on the Number Six, John Baez
- 6 is the only even perfect pronic number
- 6 is the largest integer to be both a factorial and a primorial
- 6 is the only perfect number that is also a product of its proper divisors (submitted by Michael W. Ecker)
- 6 is the largest square-free factorial
- 6 is the only perfect factorial (submitted by Alexey Radul)
- 6 is the number of convex regular polychora in 4D space (submitted by Carlo Séquin)
- 6 is the largest triangular number whose square is also triangular
- A tetrahedron has 6 edges, a cube has 6 faces, an octahedron has 6 vertices
- The maximum number of circles of same size that fit tangentially around a given circle is 6 (submitted by Chuck DeCarlucci)
- The maximum number of side of a regular polygon that can tile a plane is 6 (submitted by Chuck DeCarlucci)
- Suppose edges of a complete graph are colored with 2 colors. Then if the number of vertices is at least 6, such graph always contains a monochromatic triangle
- Given a number of points on a plane, we connect each point to another point that is closest to it. The resulting graph has the highest degree not exceeding 6