# 12

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## Number

$12$ (**twelve**) is:

- $2^2 \times 3$

- The base of the duodecimal number system

- The number of edges of the regular octahedron and its dual, the cube

- The number of distinct pentominoes, up to reflection

- The $1$st power of $12$ after the zeroth $1$:
- $12 = 12^1$

- The $1$st of three $2$-digit integers divisible by both the sum and product of its digits:
- $12 = \paren {1 + 2} \times 4 = \paren {1 \times 2} \times 6$

- The $1$st abundant number:
- $1 + 2 + 3 + 4 + 6 = 16 > 12$

- The $2$nd semiperfect number after $6$:
- $12 = 2 + 4 + 6$

- The $3$rd pentagonal number after $1$, $5$:
- $12 = 1 + 4 + 7 = \dfrac {3 \paren {3 \times 3 - 1} } 2$

- The $3$rd superfactorial after $1$, $2$:
- $12 = 3\$ = 3! \times 2! \times 1!$

- The $4$th special highly composite number after $1$, $2$, $6$

- The $4$th of $6$ integers after $2$, $5$, $8$ which cannot be expressed as the sum of distinct triangular numbers

- The $5$th highly composite number after $1$, $2$, $4$, $6$:
- $\map {\sigma_0} {12} = 6$

- The $5$th superabundant number after $1$, $2$, $4$, $6$, and the smallest which is also abundant:
- $\dfrac {\map {\sigma_1} {12} } {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$

- The $5$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
- $3$, $4$, $6$, $7$, $12$

- The $5$th generalized pentagonal number after $1$, $2$, $5$, $7$:
- $12 = \dfrac {3 \paren {3 \times 3 - 1} } 2$

- The $5$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $\ldots$

- The number of distinct free pentominoes

- The $6$th even number after $2$, $4$, $6$, $8$, $10$ which cannot be expressed as the sum of $2$ composite odd numbers

- The $7$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$ such that all smaller positive integers coprime to it are prime

- The $7$th positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $6$, $7$, $8$, $11$, $12$, $\ldots$

- The $8$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$:
- $\map {\sigma_1} {12} = 28$

- The $8$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

- The $9$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

- The $10$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

- The $11$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$:
- $12 = 6 \times 2 = 6 \times \paren {1 \times 2}$

- The $11$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$:
- $12 = 4 \times 3 = 4 \times \paren {1 + 2}$

- The square of the reversal of $12$ equals the reversal of the square of $12$:
- $12^2 = 144$
- $21^2 = 441$

- $12 = 3 \times 4$, and $56 = 7 \times 8$

## Also see

- 12 Pentominoes
- Product of Proper Divisors of 12
- 12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14
- 8 Mutually Non-Attacking Queens on Chessboard
- 12 Knights to Attack or Occupy All Squares on Chessboard
- Twelve Factorial plus One is divisible by 13 Squared
- Square of Reversal of Small-Digit Number

*Previous ... Next*: Superfactorial

*Previous ... Next*: Semiperfect Number*Previous ... Next*: Highly Composite Number*Previous ... Next*: Superabundant Number*Previous ... Next*: Special Highly Composite Number

*Previous ... Next*: Sequence of Integers whose Factorial minus 1 is Prime*Previous ... Next*: Generalized Pentagonal Number

*Previous ... Next*: Integers such that all Coprime and Less are Prime*Previous ... Next*: Integers not Sum of Distinct Triangular Numbers

*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Harshad Number

## Historical Note

There are many occurrences of the number $12$ in various cultures in history:

- There are $12$ signs of the Zodiac, divided into $3$ sets of $4$ each, and categorising them another way, into $4$ sets of $3$ each.

- There are $12$ hours in each day and in each night.

- There are $12$ semitones in an octave of a well-tempered scale.

- There were $12$ pence in $1$ shilling in pre-decimal British coinage.

The word dozen, now falling into disuse, means a set of $12$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $12$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $12$

Categories:

- Powers of 12/Examples
- Superfactorials/Examples
- Pentagonal Numbers/Examples
- Semiperfect Numbers/Examples
- Highly Composite Numbers/Examples
- Superabundant Numbers/Examples
- Special Highly Composite Numbers/Examples
- Generalized Pentagonal Numbers/Examples
- Highly Abundant Numbers/Examples
- Harshad Numbers/Examples
- Zuckerman Numbers/Examples
- Abundant Numbers/Examples
- Specific Numbers
- 12