# 216 (number)

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

Cardinal | two hundred sixteen | |||

Ordinal | 216th (two hundred sixteenth) | |||

Factorization | 2^{3} × 3^{3} | |||

Divisors | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 | |||

Greek numeral | ΣΙϚ´ | |||

Roman numeral | CCXVI | |||

Binary | 11011000_{2} | |||

Ternary | 22000_{3} | |||

Quaternary | 3120_{4} | |||

Quinary | 1331_{5} | |||

Senary | 1000_{6} | |||

Octal | 330_{8} | |||

Duodecimal | 160_{12} | |||

Hexadecimal | D8_{16} | |||

Vigesimal | AG_{20} | |||

Base 36 | 60_{36} |

**216** (**two hundred [and] sixteen**) is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato.

## In mathematics

216 is the cube of 6, and the sum of three cubes:[math]\displaystyle{ 216=6^3=3^3+4^3+5^3. }[/math]
It is the smallest cube that can be represented as a sum of three positive cubes,^{[1]} making it the first nontrivial example for Euler's sum of powers conjecture. It is, moreover, the smallest number that can be represented as a sum of any number of distinct positive cubes in more than one way.^{[2]} It is a highly powerful number: the product [math]\displaystyle{ 3\times 3 }[/math] of the exponents in its prime factorization [math]\displaystyle{ 216 = 2^3\times 3^3 }[/math] is larger than the product of exponents of any smaller number.^{[3]}

Because there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number.^{[4]} Although it is not a semiprime, the three closest numbers on either side of it are, making it the middle number between twin semiprime-triples, the smallest number with this property.^{[5]} Sun Zhiwei has conjectured that each natural number not equal to 216 can be written as either a triangular number or as a triangular number plus a prime number; however, this is not possible for 216. If the conjecture is true, 216 would be the only number for which this is not possible.^{[6]}

There are 216 ordered pairs of four-element permutations whose products generate all the other permutations on four elements.^{[7]} There are also 216 fixed hexominoes, the polyominoes made from 6 squares, joined edge-to-edge. Here "fixed" means that rotations or mirror reflections of hexominoes are considered to be distinct shapes.^{[8]}

## In other fields

216 is one common interpretation of Plato's number, a number described in vague terms by Plato in the *Republic*. Other interpretations include 3600 and 12960000.^{[9]}

There are 216 colors in the web-safe color palette, a [math]\displaystyle{ 6\times 6\times 6 }[/math] color cube.^{[10]}

In the game of checkers, there are 216 different positions that can be reached by the first three moves.^{[11]}

## See also

- The year 216
- List of highways numbered 216
- All pages with titles containing
*216*

## References

- ↑ Sloane, N. J. A., ed. "Sequence A066890 (Cubes that are the sum of three distinct positive cubes)". OEIS Foundation. https://oeis.org/A066890.
- ↑ Sloane, N. J. A., ed. "Sequence A003998 (Numbers that are a sum of distinct positive cubes in more than one way)". OEIS Foundation. https://oeis.org/A003998.
- ↑ Sloane, N. J. A., ed. "Sequence A005934 (Highly powerful numbers)". OEIS Foundation. https://oeis.org/A005934.
- ↑ Sloane, N. J. A., ed. "Sequence A005114 (Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function)". OEIS Foundation. https://oeis.org/A005114.
- ↑ Sloane, N. J. A., ed. "Sequence A202319 (Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple)". OEIS Foundation. https://oeis.org/A202319.
- ↑ "On sums of primes and triangular numbers".
*Journal of Combinatorics and Number Theory***1**(1): 65–76. 2009. - ↑ Sloane, N. J. A., ed. "Sequence A071605 (Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n)". OEIS Foundation. https://oeis.org/A071605.
- ↑ Sloane, N. J. A., ed. "Sequence A001168 (Number of fixed polyominoes with n cells)". OEIS Foundation. https://oeis.org/A001168.
- ↑ Adam, J. (February 1902). "The arithmetical solution of Plato's number".
*The Classical Review***16**(1): 17–23. - ↑ Thomas, B. (1998). "Palette's plunder".
*IEEE Internet Computing***2**(2): 87–89. doi:10.1109/4236.670691. - ↑ Sloane, N. J. A., ed. "Sequence A133047 (Starting from the standard 12 against 12 starting position in checkers, the sequence gives the number of distinct positions that can arise after n moves)". OEIS Foundation. https://oeis.org/A133047.

Original source: https://en.wikipedia.org/wiki/216 (number).
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