# 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 12

- 12 is the smallest abundant number
- 12 is the kissing number in three dimensions
- There are 12 pentominos, the polyominoes made from 5 squares
- 12 is the only number such that it is equal to the sum of 3 raised to its digits: 12 = 3
^{1} + 3^{2}
- 12 is the smallest number such that it is equal to the sum of its digits plus the cubes of its digits: 12 = 1 + 2 + 1
^{3} + 2^{3}
- Cube has 12 edges, dodecahedron has 12 faces, icosahedron has 12 vertices
- 12 is the weight of the modular discriminant as a modular form (submitted by Qiaochu Yuan)

## Rare Properties of 12

The number is called *pronic* if it is the product of two consecutive numbers.

They are twice triangular numbers.

## Common Properties of 12

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

A number is *even* if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

The number n is *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.