216 (number)

← 215 216 217 →
← 210 211 212 213 214 215 216 217 218 219 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	two hundred sixteen
Ordinal	216th (two hundred sixteenth)
Factorization	23 × 33
Divisors	1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Greek numeral	ΣΙϚ´
Roman numeral	CCXVI
Binary	110110002
Ternary	220003
Senary	10006
Octal	3308
Duodecimal	16012
Hexadecimal	D816

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

Visual proof that 3 + 4 + 5 = 6

216 is the cube of 6, and the sum of three cubes:

$${\displaystyle 216=6^{3}=3^{3}+4^{3}+5^{3}.}$$

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 ${\displaystyle 3\times 3}$ of the exponents in its prime factorization ${\displaystyle 216=2^{3}\times 3^{3}}$ 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 ${\displaystyle 6\times 6\times 6}$ color cube. ^[10]

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

The proto-Kabbalistic work Sefer Yetzirah states that the creation of the world was achieved by the manipulation of 216 sacred letters. ^[12]

See also

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

References

1. Sloane, N. J. A. (ed.). "SequenceA066890(Cubes that are the sum of three distinct positive cubes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA003998(Numbers that are a sum of distinct positive cubes in more than one way)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Sloane, N. J. A. (ed.). "SequenceA005934(Highly powerful numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. Sloane, N. J. A. (ed.). "SequenceA005114(Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA202319(Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. Sun, Zhi-Wei (2009). "On sums of primes and triangular numbers". Journal of Combinatorics and Number Theory. 1 (1): 65–76. arXiv: 0803.3737 . MR   2681507.
7. Sloane, N. J. A. (ed.). "SequenceA071605(Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA001168(Number of fixed polyominoes with n cells)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Adam, J. (February 1902). "The arithmetical solution of Plato's number". The Classical Review. 16 (1): 17–23. doi:10.1017/S0009840X0020526X. JSTOR   694295. S2CID   161664478.
10. Thomas, B. (1998). "Palette's plunder". IEEE Internet Computing. 2 (2): 87–89. doi:10.1109/4236.670691.
11. Sloane, N. J. A. (ed.). "SequenceA133047(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)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
12. Encyclopaedia Judaica, 2nd ed., vol. VI, Keter Publishing House, p. 232