111 (number) explained

Number:	111
Divisor:	1, 3, 37, 111

111 (one hundred [and] eleven) is the natural number following 110 and preceding 112.

In mathematics

111 is the fourth non-trivial nonagonal number,^[1] and the seventh perfect totient number.^[2]

\varphi(n)

of 72 is equal to the totient value of its sum-of-divisors:

\varphi(111)=\varphi(\sigma(111)).

^[3]

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively).

Magic squares

The smallest magic square using only 1 and prime numbers has a magic constant of 111:^[4]

31	73	7
13	37	61
67	1	43

Also, a six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:

1	11	31	29	19	20
2	22	24	25	8	30
3	33	26	23	17	9
34	27	10	12	21	7
35	14	15	16	18	13
36	4	5	6	28	32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

On the other hand, 111 lies between 110 and 112, which are the two smallest edge-lengths of squares that are tiled in the interior by smaller squares of distinct edge-lengths (see, squaring the square).^[5]

Properties in certain radices

111 is

R₃

or the second repunit in decimal, a number like 11, 111, or 1111 that consists of repeated units, or ones. 111 equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form

3n x 37

. As a repunit, it also follows that 111 is a palindromic number. All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in decimal, but is prime in base two, where 111₂ = 7₁₀. It is also prime in many other bases up to 128 (3, 5, 6, ..., 119) . In base 10, it is furthermore a strobogrammatic number,^[6] as well as a Harshad number.^[7]

In base 18, the number 111 is 7³ (= 343₁₀) which is the only base where 111 is a perfect power.

Nelson

See main article: Nelson (cricket). In cricket, the number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate—Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole".

Particularly in cricket, multiples of 111 are called a double Nelson (222), triple Nelson (333), quadruple Nelson (444; also known as a salamander) and so on.

A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.

In other fields

111 is also:

The atomic number of the element roentgenium (Rg).
The chemical compound 1,1,1-trichloroethane is a chlorinated hydrocarbon that was used as an industrial solvent with a trade name "Solvent 111".
The emergency telephone number in New Zealand; see 111 (emergency telephone number).
NHS 111, a medical helpline in England and Scotland.
It is the lowest positive integer requiring seven syllables to name in English (British and Commonwealth), or six syllables (by dropping the "and") in American English.
Occasionally it is referred to as "eleventy-one", as read in The Fellowship of the Ring by J.R.R. Tolkien.^[8]
The first Pacific (4-6-2) locomotive in Great Britain was the Great Western Railway's No. 111 The Great Bear of 1908, designed by George Jackson Churchward. This locomotive was rebuilt into a 4-6-0 Castle Class engine in 1924, renamed Viscount Churchill, and retained the fleet number 111 until scrapped in July 1953.
In Pakistani history, Pakistan Army's 111 Brigade is notorious for being used for staging the coups to overthrow the elected governments in order to impose martial law.

References

Notes and References

A001106. 9-gonal (or enneagonal or nonagonal) numbers. 2016-05-26.
A082897. Perfect totient numbers. 2016-05-26.
2024-02-03 .
Book: Henry E. Dudeney . Henry Dudeney . Amusements in Mathematics . . London . 1917 . 125 . 978-1153585316 . 645667320 .
Gambini . Ian . A method for cutting squares into distinct squares . . 98 . 1–2 . . Amsterdam . 65–80 . 1999 . 10.1016/S0166-218X(99)00158-4 . free . 1723687 . 0935.05024 .
A000787. Strobogrammatic numbers. 2022-05-07.
A005349. Niven (or Harshad) numbers. 2016-05-26.
Book: John Ronald Reuel Tolkien. The fellowship of the ring: being the first part of The lord of the rings. 1993. HarperCollins. 978-0-261-10235-4.