216 (number)

Short description: Natural number

| <- 215  | 216 |  217 ->

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← 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
Quaternary| 31204
Quinary| 13315
Senary| 10006
Octal| 3308
Duodecimal| 16012
Hexadecimal| D816
Vigesimal| AG20
Base 36| 6036

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.

## Contents

* 1 In mathematics
* 2 In other fields
* 3 See also
* 4 References

## In mathematics

Visual proof that 33 \+ 43 \+ 53 = 63

216 is the cube of 6, and the sum of three cubes:216=63=33+43+53. 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 3×3 of the exponents in its prime factorization 216=23×33 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 6×6×6 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. "Sequence A066890 (Cubes that are the sum of three distinct positive cubes)". OEIS Foundation. https://oeis.org/A066890.
2. ↑ 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.
3. ↑ Sloane, N. J. A., ed. "Sequence A005934 (Highly powerful numbers)". OEIS Foundation. https://oeis.org/A005934.
4. ↑ 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.
5. ↑ 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.
6. ↑ "On sums of primes and triangular numbers". Journal of Combinatorics and Number Theory 1 (1): 65–76. 2009.
7. ↑ 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.
8. ↑ Sloane, N. J. A., ed. "Sequence A001168 (Number of fixed polyominoes with n cells)". OEIS Foundation. https://oeis.org/A001168.
9. ↑ Adam, J. (February 1902). "The arithmetical solution of Plato's number". The Classical Review 16 (1): 17–23. doi:10.1017/S0009840X0020526X. https://zenodo.org/record/1840799.
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. "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.
12. ↑ Encyclopaedia Judaica, 2nd ed., vol. VI, Keter Publishing House, p. 232

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