222 Divine

Some Interesting Facts Associated with the Second Repunit

Compiled by Simon J. Whitechapel

• All triplets in all bases are multiples of 111 in that base, therefore 111 is the only triplet that can ever be prime. It is not prime in base ten, but is prime in these bases 2 through 100: 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, and 99. 111 is also prime in base 111.¹

• The smallest magic square using only prime numbers (and 1) has a magic constant of 111.² Note too the literally central role of 37:

• A six-by-six magic square using the numbers 1 through 36 also has a magic constant of 111:

• It is also possible to find a “natural” magic square with a magic total of 222. To see how, we need to examine some facts about reciprocals, or numbers divided into 1.

The decimal expansion of a reciprocal either ends, like 0·5 = 1/2, or repeats endlessly, like 0·142857142857... = 1/7. 1/7 repeats endlessly because the remainders fall into a loop: 10 / 7 = 1 remainder 3; 30 / 7 = 4 remainder 2; 20 / 7 = 2 remainder 6; 60 / 7 = 8 remainder 4; 40 / 7 = 5 remainder 5; 50 / 7 = 7 remainder 1; 10 / 7 = 1 remainder 3... The remainders in a reciprocal like 1/7 can take any value from one to the-divisor-minus-one, so the decimal expansion of 1/7 must repeat before or at the 6th digit after the decimal point. Similarly, the decimal expansion of 1/11 must repeat before or at the 10th digit after the decimal point, and the decimal expansion of 1/13 before or at the 12th digit after the decimal point. Reciprocals with the maximum decimal period — the number of digits before looping begins — are always reciprocals of prime numbers. The converse is not true, however: prime reciprocals do not always have maximum period, and in base ten only nine greater than 1/100 do: 1/7, 1/17, 1/19, 1/23, 1/29, 1/47, 1/59, 1/61 and 1/97. Only one of these has an additional property sometimes found in maximum-period prime reciprocals: it forms a perfect magic square. 1/19 has this property:

01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1...
02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2...
03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3...
04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4...
05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5...
06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6...
07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7...
08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8...
09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9...
10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0...
11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1...
12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2...
13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3...
14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4...
15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5...
16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6...
17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7...
18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8...

The magic constant for this square is 81, which can be derived from the formula (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 10 x 18⁷, or 9 x 18 / 2. 666 = 9 x 148 / 2, and 148+1 or 149 is a prime number whose reciprocal has maximum period. Can you make a magic square with magic constant 666 from the multiples of 1/149? Unfortunately, no, you can’t: rows and columns all add up to 666, but neither diagonal does. This “magic square” property of 19 and its reciprocal is in fact very rare: the only other prime less than 1000 possessing it in base ten is 383 (magic constant = 9 x 382 / 2 = 1719).

Note the qualification, however: in base ten. In base three, for example, the magic constant formula is (0 + 1 + 2) / 3 x period, or 1 x period. 222 = 1 x 222, and 222+1 or 223 is a prime number whose reciprocal in base three has maximum period. Can you make a magic square with magic constant 222 from the multiples of 1/223 in base three? You can: rows and columns all add up to 222, and so do both diagonals.⁸

• If numbers other than primes are used, you can find other reciprocal magic squares with a magic total of 222 and 666.

1/24245 = 0·	0	8	16	24	32	41	4	12	20	28	37
23/24245 = 0·	4	12	20	28	37	0	8	16	24	32	41
45/24245 = 0·	8	16	24	32	41	4	12	20	28	37	0
67/24245 = 0·	12	20	28	37	0	8	16	24	32	41	4
89/24245 = 0·	16	24	32	41	4	12	20	28	37	0	8
111/24245 = 0·	20	28	37	0	8	16	24	32	41	4	12
133/24245 = 0·	24	32	41	4	12	20	28	37	0	8	16
155/24245 = 0·	28	37	0	8	16	24	32	41	4	12	20
177/24245 = 0·	32	41	4	12	20	28	37	0	8	16	24
199/24245 = 0·	37	0	8	16	24	32	41	4	12	20	28
221/24245 = 0·	41	4	12	20	28	37	0	8	16	24	32

For 1/242₄₅ each row and col = 222, the l-r diagonal = 222, and the r-l diagonal = 222 (11 x 11).

1/127667 = 0·	5	168	42	10	336	84	21
2/127667 = 0·	10	336	84	21	5	168	42
4/127667 = 0·	21	5	168	42	10	336	84
8/127667 = 0·	42	10	336	84	21	5	168
16/127667 = 0·	84	21	5	168	42	10	336
32/127667 = 0·	168	42	10	336	84	21	5
64/127667 = 0·	336	84	21	5	168	42	10

For 1/127₆₆₇ each row and col = 666, the l-r diagonal = 666, and the r-l diagonal = 666 (7 x 7). Note that each diagonal contains the same digits as each row and column (the digits in the first row are 5, 168, 42, 10, 336, 84, 21... and the pattern in the l-r diagonal is 5, skip 3, 336, skip 3, 168, skip 3, 84...).

• A magic star with 55 vertices has a magic total of 222:

(110,109,  1,  2),(  6,  1,108,107),(  2,108,104,  8),(107,104,  7,  4),
(  8,  7,106,101),(  4,106,103,  9),(101,103, 15,  3),(  9, 15,105, 93),
(  3,105,102, 12),( 93,102, 22,  5),( 12, 22,100, 88),(  5,100, 99, 18),
( 88, 99, 25, 10),( 18, 25, 98, 81),( 10, 98, 97, 17),( 81, 97, 33, 11),
( 17, 33, 96, 76),( 11, 96, 95, 20),( 76, 95, 38, 13),( 20, 38, 94, 70),
( 13, 94, 92, 23),( 70, 92, 46, 14),( 23, 46, 91, 62),( 14, 91, 90, 27),
( 62, 90, 54, 16),( 27, 54, 89, 52),( 16, 89, 87, 30),( 52, 87, 64, 19),
( 30, 64, 86, 42),( 19, 86, 85, 32),( 42, 85, 74, 21),( 32, 74, 82, 34),
( 21, 82, 84, 35),( 34, 84, 80, 24),( 35, 80, 79, 28),( 24, 79, 83, 36),
( 28, 83, 72, 39),( 36, 72, 77, 37),( 39, 77, 75, 31),( 37, 75, 69, 41),
( 31, 69, 78, 44),( 41, 78, 58, 45),( 44, 58, 65, 55),( 45, 65, 51, 61),
( 55, 51, 48, 68),( 61, 48, 57, 56),( 68, 57, 47, 50),( 56, 47, 53, 66),
( 50, 53, 59, 60),( 66, 59, 26, 71),( 60, 26, 73, 63),( 71, 73, 49, 29),
( 63, 49, 43, 67),( 29, 43, 40,110),( 67, 40,109,  6).

Outer numbers=

110,  6,  2,107,  8,  4,101,  9,  3, 93,
12,  5, 88, 18, 10, 81, 17, 11, 76, 20,
13, 70, 23, 14, 62, 27, 16, 52, 30, 19,
42, 32, 21, 34, 35, 24, 28, 36, 39, 37,
31, 41, 44, 45, 55, 61, 68, 56, 50, 66,
60, 71, 63, 29, 67.

Inner numbers=

109,  1,108,104,  7,106,103, 15,105,102,
22,100, 99, 25, 98, 97, 33, 96, 95, 38,
94, 92, 46, 91, 90, 54, 89, 87, 64, 86,
85, 74, 82, 84, 80, 79, 83, 72, 77, 75,
69, 78, 58, 65, 51, 48, 57, 47, 53, 59,
26, 73, 49, 43, 40.

And a magic star with 166 vertices has a magic total of 666: see Polygrammagick.

• Magic chains are related to magic stars but are much easier to find. In this six-link magic chain, every set of four links sums to 26:

In this 166-link magic chain, every set of four links sums to 666:

(  2,331,  1,332), (  1,332,  3,330), (  3,330,  4,329), (  4,329,  5,328),
(  5,328,  6,327), (  6,327,  7,326), (  7,326,  8,325), (  8,325,  9,324),
(  9,324, 10,323), ( 10,323, 11,322), ( 11,322, 12,321), ( 12,321, 13,320),
( 13,320, 14,319), ( 14,319, 15,318), ( 15,318, 16,317), ( 16,317, 17,316),
( 17,316, 18,315), ( 18,315, 19,314), ( 19,314, 20,313), ( 20,313, 21,312),
( 21,312, 22,311), ( 22,311, 23,310), ( 23,310, 24,309), ( 24,309, 25,308),
( 25,308, 26,307), ( 26,307, 27,306), ( 27,306, 28,305), ( 28,305, 29,304),
( 29,304, 30,303), ( 30,303, 31,302), ( 31,302, 32,301), ( 32,301, 33,300),
( 33,300, 34,299), ( 34,299, 35,298), ( 35,298, 36,297), ( 36,297, 37,296),
( 37,296, 38,295), ( 38,295, 39,294), ( 39,294, 40,293), ( 40,293, 41,292),
( 41,292, 42,291), ( 42,291, 43,290), ( 43,290, 44,289), ( 44,289, 45,288),
( 45,288, 46,287), ( 46,287, 47,286), ( 47,286, 48,285), ( 48,285, 49,284),
( 49,284, 50,283), ( 50,283, 51,282), ( 51,282, 52,281), ( 52,281, 53,280),
( 53,280, 54,279), ( 54,279, 55,278), ( 55,278, 56,277), ( 56,277, 57,276),
( 57,276, 58,275), ( 58,275, 59,274), ( 59,274, 60,273), ( 60,273, 61,272),
( 61,272, 62,271), ( 62,271, 63,270), ( 63,270, 64,269), ( 64,269, 65,268),
( 65,268, 66,267), ( 66,267, 67,266), ( 67,266, 68,265), ( 68,265, 69,264),
( 69,264, 70,263), ( 70,263, 71,262), ( 71,262, 72,261), ( 72,261, 73,260),
( 73,260, 74,259), ( 74,259, 75,258), ( 75,258, 76,257), ( 76,257, 77,256),
( 77,256, 78,255), ( 78,255, 79,254), ( 79,254, 80,253), ( 80,253, 81,252),
( 81,252, 82,251), ( 82,251, 83,250), ( 83,250, 84,249), ( 84,249, 85,248),
( 85,248, 86,247), ( 86,247, 87,246), ( 87,246, 88,245), ( 88,245, 89,244),
( 89,244, 90,243), ( 90,243, 91,242), ( 91,242, 92,241), ( 92,241, 93,240),
( 93,240, 94,239), ( 94,239, 95,238), ( 95,238, 96,237), ( 96,237, 97,236),
( 97,236, 98,235), ( 98,235, 99,234), ( 99,234,100,233), (100,233,101,232),
(101,232,102,231), (102,231,103,230), (103,230,104,229), (104,229,105,228),
(105,228,106,227), (106,227,107,226), (107,226,108,225), (108,225,109,224),
(109,224,110,223), (110,223,111,222), (111,222,112,221), (112,221,113,220),
(113,220,114,219), (114,219,115,218), (115,218,116,217), (116,217,117,216),
(117,216,118,215), (118,215,119,214), (119,214,120,213), (120,213,121,212),
(121,212,122,211), (122,211,123,210), (123,210,124,209), (124,209,125,208),
(125,208,126,207), (126,207,127,206), (127,206,128,205), (128,205,129,204),
(129,204,130,203), (130,203,131,202), (131,202,132,201), (132,201,133,200),
(133,200,134,199), (134,199,135,198), (135,198,136,197), (136,197,137,196),
(137,196,138,195), (138,195,139,194), (139,194,140,193), (140,193,141,192),
(141,192,142,191), (142,191,143,190), (143,190,144,189), (144,189,145,188),
(145,188,146,187), (146,187,147,186), (147,186,148,185), (148,185,149,184),
(149,184,150,183), (150,183,151,182), (151,182,152,181), (152,181,153,180),
(153,180,154,179), (154,179,155,178), (155,178,156,177), (156,177,157,176),
(157,176,158,175), (158,175,159,174), (159,174,160,173), (160,173,161,172),
(161,172,162,171), (162,171,163,170), (163,170,165,168), (165,168,164,169),
(164,169,166,167), (166,167,  2,331).

• The most obvious triplet in the Bible is, of course, 666, or the number of the Beast in Revelation xiii, 18:

Here is wisdom: let him who hath understanding count of the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.

                         r     s    q     n   w    r     n

666 has a number of curious properties. For example:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 = 666.

• In the alphabetic Hebrew counting system, 222 is RKB, which, pointed as ReKeB, means “chariot”. In the Hebrew Bible, RKB appears in plural form in the 2nd chapter of Haggai, but it is in the 23rd, not the 22nd verse. The Hebrew Bible was translated into Greek by Jewish scholars about 2,200 years ago. In this translation, called the Septuagint, Haggai’s chariots still appear in the 23rd verse:

22 Eipon pros Zorobabel ton tou Salathihl ek phyles Iouda, legon, Ego seio ton ouranon kai ten gen, kai ten thalassan kai ten kheran, 23 kai katastrepso thronous basileon, kai olothreuso dynamin basileon ton ethnon, kai katastrepso harmata kai anabatas, kai katabesontai hippoi kai anabatai auton hekatos en romphaia pros ton adelphon autou.

Next came the Vulgate, or Latin Bible, but we are not third time lucky, for Haggai’s chariots still appear in the 23rd verse:

22 Loquere ad Zorobabbel, ducem Iuda, dicens: Ego movebo caelum pariter et terram. 23 Et subvertam solium regnorum, et conteram fortitudinem regni gentium; et subvertam quadrigam et ascensorem ejus; et descendent equi, et ascensores eorum, vir in gladio fratris sui.

22 And I will overthrow the throne of kingdoms, and I will destroy the strength of the kingdoms of the heathen; and I will overthrow the chariots, and those that ride in them; and the horses and their riders shall come down, every one by the sword of his brother.

•That kind of scriptural manipulation is also found in an ingenious theory that William Shakespeare had a hand in the translation of at least the 46th Psalm of the King James Bible, which runs thus:

1 God¹ is² our³ refuge⁴ and⁵ strength⁶, a⁷ very⁸ present⁹ help¹⁰ in¹¹ trouble¹².

2 Therefore¹³ will¹⁴ not¹⁵ we¹⁶ fear¹⁷, though¹⁸ the¹⁹ earth²⁰ be²¹ removed²², and²³ though²⁴ the²⁵ mountains²⁶ be²⁷ carried²⁸ into²⁹ the³⁰ midst³¹ of³² the³³ sea³⁴;

3 Though³⁵ the³⁶ waters³⁷ thereof³⁸ roar³⁹ and⁴⁰ be⁴¹ troubled⁴², though⁴³ the⁴⁴ mountains⁴⁵ shake⁴⁶ with¹ the² swelling³ thereof⁴. Selah⁵.

4 There⁶ is⁷ a⁸ river⁹, the¹⁰ streams¹¹ whereof¹² shall¹³ make¹⁴ glad¹⁵ the¹⁶ city¹⁷ of¹⁸ God¹⁹, the²⁰ holy²¹ place²² of²³ the²⁴ tabernacles²⁵ of²⁶ the²⁷ most²⁸ High²⁹.

5 God³⁰ is³¹ in³² the³³ midst³⁴ of³⁵ her³⁶; she³⁷ shall³⁸ not³⁹ be⁴⁰ moved⁴¹: God⁴² shall⁴³ help⁴⁴ her⁴⁵, and⁴⁶ that⁴⁷ right⁴⁸ early⁴⁹.

6 The⁵⁰ heathen⁵¹ raged⁵², the⁵³ kingdoms⁵⁴ were⁵⁵ moved⁵⁶: he⁵⁷ uttered⁵⁸ his⁵⁹ voice⁶⁰, the⁶¹ earth⁶² melted⁶³.

7 The⁶⁴ LORD⁶⁵ of⁶⁶ hosts⁶⁷ is⁶⁸ with⁶⁹ us⁷⁰; the⁷¹ God⁷² of⁷³ Jacob⁷⁴ is⁷⁵ our⁷⁶ refuge⁷⁷. Selah⁷⁸.

8 Come⁷⁹, behold⁸⁰ the⁸¹ works⁸² of⁸³ the⁸⁴ LORD⁸⁵, what⁸⁶ desolations⁸⁷ he⁸⁸ hath⁸⁹ made⁹⁰ in⁹¹ the⁹² earth⁹³.

9 He⁹⁴ maketh⁹⁵ wars⁹⁶ to⁹⁷ cease⁹⁸ unto⁹⁹ the¹⁰⁰ end¹⁰¹ of¹⁰² the¹⁰³ earth¹⁰⁴; he¹⁰⁵ breaketh¹⁰⁶ the¹⁰⁷ bow¹⁰⁸, and¹⁰⁹ cutteth¹¹⁰ the¹¹¹ spear⁴⁶ in⁴⁵ sunder⁴⁴; he⁴³ burneth⁴² the⁴¹ chariot⁴⁰ in³⁹ the³⁸ fire³⁷.

10 Be³⁶ still³⁵, and³⁴ know³³ that³² I³¹ am³⁰ God²⁹: I²⁸ will²⁷ be²⁶ exalted²⁵ among²⁴ the²³
        heathen²², I²¹ will²⁰ be¹⁹ exalted¹⁸ in¹⁷ the¹⁶ earth¹⁵.

11 The¹⁴ LORD¹³ of¹² hosts¹¹ is¹⁰ with⁹ us⁸; the⁷ God⁶ of⁵ Jacob⁴ is³ our² refuge¹.
        Selah⁽⁰⁾.

• The emperor Heliogabalus, high-priest of the charioteer sun-god Elah-Gabal, died in 222 AD.

• Pope Callistus I also died in 222. Pope Vigilius died in 555, John XVI in 999, and the anti-pope Sylvester IV in 1111. Marcellus II was elected and died in the same year, 1555.

• The great German mathematician Karl Friedrich Gauss was born in 1777.

• The American Declaration of Independence was made in 1776, or 2 x 2 x 2 x 222 years after the birth of Christ in standard Christian chronology — which is certainly faulty, and may be deliberately so.⁵

• Sherlock Holmes lives at 221B Baker Street. Why the “B”? Perhaps because B or “b” is 2 in the alphabetic Greek counting system, and “a” or A is 1. So change B to 2 and 1 to A, swap them around, and Sherlock Holmes lives at 222A Baker Street.

• The isotope of the radioactive noble gas Radon with the longest half-life is 222R.⁶

• Chuck Berry’s “Thirty Days”, first recorded 21/5/55, lasts 2’22”.⁹

• If you choose one of 1 through 100 at random over and over again, the chances that you will NOT choose a number already chosen reduce according to the formula (100! / (100-n)!) / 100ⁿ, where n equals the number of times you have chosen after the first occasion. The chances of NOT having duplicated a number on the 94th, 95th, 96th, and 97th choices after the first are, respectively:

7·77718462 x 10^-32%
3·88859231 x 10^-34%
1·55543692 x 10^-35%
4·66631077 x 10^-37%

• The 1975 Penguin paperback of Evelyn Waugh’s Put Out More Flags (1942) finishes on page 222.

• A better rational approximation for pi than the traditional 22/7 is 333/106:

• However, the first triplet in the digits of p is not 333 but 111 (followed by 555):

3·14159265358979323846264338327950288419716939937510
  58209749445923078164062862089986280348253421170679
  82148086513282306647093844609550582231725359408128
  48111745028410270193852110555964462294895493038196...

(Each block after the decimal point contains 50 digits)

• The first triplet in the square root of 222, on the other hand, is 222:

√222 =

14·89966442575133971933181604612395114023452166218124
   73380574030119289350747022456370098357196526519656
   52139396484012226637696242316850851449289924318610...

• Jack the Ripper began killing women in 1888.¹⁰

• 222 is the first triplet in the square roots of 1888 and 1999, the 111th anniversary of the Ripper murders:

√1888 =

43·45112196480086289556597349894964480490664793369900
   61050597222776432794391794775715951607332203886595...

√1999 =

44·71017781221631419961342300204768423151749410304791
   57894467222633780055246726276812270371806011528674...
 

• 222 is also the first triplet in the square roots of the reciprocals of 1888 and 1999:

 √(1/1888) =

0·02301436544745808416078706223461315932463275843945
  92193353070562911246183470230283747855724222565617...

 √(1/1999) =

0·02236627204212922171066204252228498460806277844074
  43300597532377505642849047887081947108740273142335...

• The painter Sir Lawrence Alma-Tadema (1836-1912) completed a painting called The Roses of Heliogabalus in 1888.

• Alma-Tadema was 52 in 1888. The first two non-zero digits of the reciprocal of 1888 are 5 and 2. The painting is 52” high (by 84⅛” wide, encoding the golden ratio):

 1/1888 =

0·00052*966101694915254237288135593220338983050847457
  6271186440677* (repeating over 58 digits)

• The reciprocal of 1999 repeats over 999 digits and contains all the triplets except 222 and 999. Like the reciprocal of 1888, its first two non-zero digits are 5 and 2:

1/1999 =

0·*00050025012506253126563281640820410205102551275637
   81890945472736368184092046023011505752876438219109
   55477738869434717358679339669834917458729364682341
   17058529264632316158079039519759879939969984992496
   24812406203101550775387693846923461730865432716358
   17908954477238619309654827413706853426713356678339
   16958479239619809904952476238119059529764882441220
   61030515257628814407203601800900450225112556278139
   06953476738369184592296148074037018509254627313656
   82841420710355177588794397198599299649824912456228
   11405702851425712856428214107053526763381690845422
   71135567783891945972986493246623311655827913956978
   48924462231115557778889444722361180590295147573786
   89344672336168084042021010505252626313156578289144
   57228614307153576788394197098549274637318659329664
   83241620810405202601300650325162581290645322661330
   66533266633316658329164582291145572786393196598299
   14957478739369684842421210605302651325662831415707
   85392696348174087043521760880440220110055027513756
   8784392196098049024512256128064032016008004002001* (repeating over 999 digits)
 
• Radii at 0º and 222º divide a circle in the golden ratio,¹¹ which is encoded by the dimensions of The Roses of Heliogabalus: 84⅛” x 52” (84·125 / 52 = 1·617788462...; 52 / 84·125 = 0·618127786...).

• The golden ratio is also known as f or phi and is equal to (√5 + 1) / 2 or 1/((√5 + 1) / 2) = 1·61803398874989... or 0·61803398874989...

f x 2p =  3·8832220774...
f x 4p =  7·7664441549...
f x 8p = 15·5328883098...

f x 2^{2^2} (16)     =  9·88854382...
f x 2^{2^2} x 2 (32) = 19·77708764...

f² x 2^{2^2} (16)     =  6·11145618...
f² x 2^{2^2} x 2 (32) = 12·22291236...

f x 13  =   8·0344418537...
f x 13² = 104·44774409...

• The first thousand (and one) digits of f are:

1·61803398874989484820458683436563811772030917980576
  28621354486227052604628189024497072072041893911374
  84754088075386891752126633862223536931793180060766
  72635443338908659593958290563832266131992829026788
  06752087668925017116962070322210432162695486262963
  13614438149758701220340805887954454749246185695364
  86444924104432077134494704956584678850987433944221
  25448770664780915884607499887124007652170575179788
  34166256249407589069704000281210427621771117778053
  15317141011704666599146697987317613560067087480710
  13179523689427521948435305678300228785699782977834
  78458782289110976250030269615617002504643382437764
  86102838312683303724292675263116533924731671112115
  88186385133162038400522216579128667529465490681131
  71599343235973494985090409476213222981017261070596
  11645629909816290555208524790352406020172799747175
  34277759277862561943208275051312181562855122248093
  94712341451702237358057727861600868838295230459264
  78780178899219902707769038953219681986151437803149
  97411069260886742962267575605231727775203536139362...

The digits of an irrational numbers like phi are believed to be perfectly random, and each digit should therefore occur with a frequency of 1/10, each two-digit combination (00 to 99) with a frequency of 1/100, and each three-digit combination (000 to 999) with a frequency of 1/1000. 222 would therefore be expected to occur approximately once in the first 1000 digits of phi but in fact occurs five times. The odds of five or more such occurences are about 1 in thirty-nine thousand, six hundred and fifty-nine.¹²

• The first triplet in e, “the base of natural logarithms”,¹³ is 999:

  2·71828182845904523536028747135266249775724709369995...

• j x e = 1·6799905609...

• An aliquot sequence is the sequence formed by taking the sum of the divisors of n less than n (including 1), then repeating the process until it reaches 1 (or forms a loop, as with 220, whose aliquot sum is 284, and 284, whose aliquot sum is 220). 222 has one of the longest aliquot sequences of any number below 1000.

215₁ > 49₂ > 8₃ > 7₄ > 1₅.

216₁ > 384₂ > 636₃ > 876₄ > 1196₅ > 1156₆ > 993₇ > 335₈ > 73₉ > 1₁₀.

217₁ > 39₂ > 17₃ > 1₄.

218₁ > 112₂ > 136₃ > 134₄ > 70₅ > 74₆ > 40₇ > 50₈ > 43₉ > 1₁₀.

219₁ > 77₂ > 19₃ > 1₄.

220₁ > 284₂ > 220₃.

221₁ > 31₂ > 1₃.

222₁ > 234₂ > 312₃ > 528₄ > 960₅ > 2088₆ > 3762₇ > 5598₈ > 6570₉ > 10746₁₀ > 13254₁₁ > 13830₁₂ > 19434₁₃ > 20886₁₄ > 21606₁₅ > 25098₁₆ > 26742₁₇ > 26754₁₈ > 40446₁₉ > 63234₂₀ > 77406₂₁ > 110754₂₂ > 171486₂₃ > 253458₂₄ > 295740₂₅ > 647748₂₆ > 1077612₂₇ > 1467588₂₈ > 1956812₂₉ > 2109796₃₀ > 1889486₃₁ > 953914₃₂ > 668966₃₃ > 353578₃₄ > 176792₃₅ > 254128₃₆ > 308832₃₇ > 502104₃₈ > 753216₃₉ > 1240176₄₀ > 2422288₄₁ > 2697920₄₂ > 3727264₄₃ > 3655076₄₄ > 2760844₄₅ > 2100740₄₆ > 2310856₄₇ > 2455544₄₈ > 3212776₄₉ > 3751064₅₀ > 3282196₅₁ > 2723020₅₂ > 3035684₅₃ > 2299240₅₄ > 2988440₅₅ > 5297320₅₆ > 8325080₅₇ > 11222920₅₈ > 15359480₅₉ > 19199440₆₀ > 28875608₆₁ > 25266172₆₂ > 19406148₆₃ > 26552604₆₄ > 40541052₆₅ > 54202884₆₆ > 72270540₆₇ > 147793668₆₈ > 228408732₆₉ > 348957876₇₀ > 508132204₇₁ > 404465636₇₂ > 303708376₇₃ > 290504024₇₄ > 312058216₇₅ > 294959384₇₆ > 290622016₇₇ > 286081174₇₈ > 151737434₇₉ > 75868720₈₀ > 108199856₈₁ > 101437396₈₂ > 76247552₈₃ > 76099654₈₄ > 42387146₈₅ > 21679318₈₆ > 12752594₈₇ > 7278382₈₈ > 3660794₈₉ > 1855066₉₀ > 927536₉₁ > 932464₉₂ > 1013592₉₃ > 1546008₉₄ > 2425752₉₅ > 5084088₉₆ > 8436192₉₇ > 13709064₉₈ > 20563656₉₉ > 33082104₁₀₀ > 57142536₁₀₁ > 99483384₁₀₂ > 245978376₁₀₃ > 487384824₁₀₄ > 745600776₁₀₅ > 1118401224₁₀₆ > 1677601896₁₀₇ > 2538372504₁₀₈ > 4119772776₁₀₉ > 8030724504₁₁₀ > 14097017496₁₁₁ > 21148436904₁₁₂ > 40381357656₁₁₃ > 60572036544₁₁₄ > 100039354704₁₁₅ > 179931895322₁₁₆ > 94685963278₁₁₇ > 51399021218₁₁₈ > 28358080762₁₁₉ > 18046051430₁₂₀ > 17396081338₁₂₁ > 8698040672₁₂₂ > 8426226964₁₂₃ > 6319670230₁₂₄ > 5422685354₁₂₅ > 3217383766₁₂₆ > 1739126474₁₂₇ > 996366646₁₂₈ > 636221402₁₂₉ > 318217798₁₃₀ > 195756362₁₃₁ > 101900794₁₃₂ > 54202694₁₃₃ > 49799866₁₃₄ > 24930374₁₃₅ > 17971642₁₃₆ > 11130830₁₃₇ > 8904682₁₃₈ > 4913018₁₃₉ > 3126502₁₄₀ > 1574810₁₄₁ > 1473382₁₄₂ > 736694₁₄₃ > 541162₁₄₄ > 312470₁₄₅ > 249994₁₄₆ > 127286₁₄₇ > 69898₁₄₈ > 34952₁₄₉ > 34708₁₅₀ > 26038₁₅₁ > 13994₁₅₂ > 7000₁₅₃ > 11720₁₅₄ > 14740₁₅₅ > 19532₁₅₆ > 16588₁₅₇ > 18692₁₅₈ > 14026₁₅₉ > 7016₁₆₀ > 6154₁₆₁ > 3674₁₆₂ > 2374₁₆₃ > 1190₁₆₄ > 1402₁₆₅ > 704₁₆₆ > 820₁₆₇ > 944₁₆₈ > 916₁₆₉ > 694₁₇₀ > 350₁₇₁ > 394₁₇₂ > 200₁₇₃ > 265₁₇₄ > 59₁₇₅ > 1₁₇₆.

223₁ > 1₂.

224₁ > 280₂ > 440₃ > 640₄ > 890₅ > 730₆ > 602₇ > 454₈ > 230₉ > 202₁₀ > 104₁₁ > 106₁₂ > 56₁₃ > 64₁₄ > 63₁₅ > 41₁₆ > 1₁₇.

225₁ > 178₂ > 92₃ > 76₄ > 64₅ > 63₆ > 41₇ > 1₈.

• And finally... there are exactly 111 ways of adding 1 to three (other) prime numbers to make 222:

1: 1 + 3 + 7 + 211 = 222
2: 1 + 3 + 19 + 199 = 222
3: 1 + 3 + 37 + 181 = 222
4: 1 + 3 + 61 + 157 = 222
5: 1 + 3 + 67 + 151 = 222
6: 1 + 3 + 79 + 139 = 222
7: 1 + 5 + 17 + 199 = 222
8: 1 + 5 + 19 + 197 = 222
9: 1 + 5 + 23 + 193 = 222
10: 1 + 5 + 37 + 179 = 222
11: 1 + 5 + 43 + 173 = 222
12: 1 + 5 + 53 + 163 = 222
13: 1 + 5 + 59 + 157 = 222
14: 1 + 5 + 67 + 149 = 222
15: 1 + 5 + 79 + 137 = 222
16: 1 + 5 + 89 + 127 = 222
17: 1 + 5 + 103 + 113 = 222
18: 1 + 5 + 107 + 109 = 222
19: 1 + 7 + 17 + 197 = 222
20: 1 + 7 + 23 + 191 = 222
21: 1 + 7 + 41 + 173 = 222
22: 1 + 7 + 47 + 167 = 222
23: 1 + 7 + 83 + 131 = 222
24: 1 + 7 + 101 + 113 = 222
25: 1 + 11 + 13 + 197 = 222
26: 1 + 11 + 17 + 193 = 222
27: 1 + 11 + 19 + 191 = 222
28: 1 + 11 + 29 + 181 = 222
29: 1 + 11 + 31 + 179 = 222
30: 1 + 11 + 37 + 173 = 222
31: 1 + 11 + 43 + 167 = 222
32: 1 + 11 + 47 + 163 = 222
33: 1 + 11 + 53 + 157 = 222
34: 1 + 11 + 59 + 151 = 222
35: 1 + 11 + 61 + 149 = 222
36: 1 + 11 + 71 + 139 = 222
37: 1 + 11 + 73 + 137 = 222
38: 1 + 11 + 79 + 131 = 222
39: 1 + 11 + 83 + 127 = 222
40: 1 + 11 + 97 + 113 = 222
41: 1 + 11 + 101 + 109 = 222
42: 1 + 11 + 103 + 107 = 222
43: 1 + 13 + 17 + 191 = 222
44: 1 + 13 + 29 + 179 = 222
45: 1 + 13 + 41 + 167 = 222
46: 1 + 13 + 59 + 149 = 222
47: 1 + 13 + 71 + 137 = 222
48: 1 + 13 + 101 + 107 = 222
49: 1 + 17 + 23 + 181 = 222
50: 1 + 17 + 31 + 173 = 222
51: 1 + 17 + 37 + 167 = 222
52: 1 + 17 + 41 + 163 = 222
53: 1 + 17 + 47 + 157 = 222
54: 1 + 17 + 53 + 151 = 222
55: 1 + 17 + 67 + 137 = 222
56: 1 + 17 + 73 + 131 = 222
57: 1 + 17 + 97 + 107 = 222
58: 1 + 17 + 101 + 103 = 222
59: 1 + 19 + 23 + 179 = 222
60: 1 + 19 + 29 + 173 = 222
61: 1 + 19 + 53 + 149 = 222
62: 1 + 19 + 71 + 131 = 222
63: 1 + 19 + 89 + 113 = 222
64: 1 + 23 + 31 + 167 = 222
65: 1 + 23 + 41 + 157 = 222
66: 1 + 23 + 47 + 151 = 222
67: 1 + 23 + 59 + 139 = 222
68: 1 + 23 + 61 + 137 = 222
69: 1 + 23 + 67 + 131 = 222
70: 1 + 23 + 71 + 127 = 222
71: 1 + 23 + 89 + 109 = 222
72: 1 + 23 + 97 + 101 = 222
73: 1 + 29 + 41 + 151 = 222
74: 1 + 29 + 43 + 149 = 222
75: 1 + 29 + 53 + 139 = 222
76: 1 + 29 + 61 + 131 = 222
77: 1 + 29 + 79 + 113 = 222
78: 1 + 29 + 83 + 109 = 222
79: 1 + 29 + 89 + 103 = 222
80: 1 + 31 + 41 + 149 = 222
81: 1 + 31 + 53 + 137 = 222
82: 1 + 31 + 59 + 131 = 222
83: 1 + 31 + 83 + 107 = 222
84: 1 + 31 + 89 + 101 = 222
85: 1 + 37 + 47 + 137 = 222
86: 1 + 37 + 53 + 131 = 222
87: 1 + 37 + 71 + 113 = 222
88: 1 + 37 + 83 + 101 = 222
89: 1 + 41 + 43 + 137 = 222
90: 1 + 41 + 53 + 127 = 222
91: 1 + 41 + 67 + 113 = 222
92: 1 + 41 + 71 + 109 = 222
93: 1 + 41 + 73 + 107 = 222
94: 1 + 41 + 79 + 101 = 222
95: 1 + 41 + 83 + 97 = 222
96: 1 + 43 + 47 + 131 = 222
97: 1 + 43 + 71 + 107 = 222
98: 1 + 47 + 61 + 113 = 222
99: 1 + 47 + 67 + 107 = 222
100: 1 + 47 + 71 + 103 = 222
101: 1 + 47 + 73 + 101 = 222
102: 1 + 53 + 59 + 109 = 222
103: 1 + 53 + 61 + 107 = 222
104: 1 + 53 + 67 + 101 = 222
105: 1 + 53 + 71 + 97 = 222
106: 1 + 53 + 79 + 89 = 222
107: 1 + 59 + 61 + 101 = 222
108: 1 + 59 + 73 + 89 = 222
109: 1 + 59 + 79 + 83 = 222
110: 1 + 61 + 71 + 89 = 222
111: 1 + 67 + 71 + 83 = 222

Base: “111” in base ten means (1 x 10²⁾ + (1 x 10¹⁾ + (1 x 10⁰), or (1 x 100) + (1 x 10) + (1 x 1). In base two, it means 1 x 2² + 1 x 2¹ + 1 x 2⁰, or (1 x 4) + (1 x 2) + (1 x 1) = 7. “222” cannot exist as such in base two (where its value in base ten would be have to be written as 11011110) but can in base three, where it means (2 x 3²) + (2 x 3¹) + (2 x 3⁰) or (2 x 9) + (2 x 3) + (2 x 1) = 36. In any base, the total number of digits is equal to the base but there is no digit with the same value as the base. In base ten, for example, there are ten digits — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -- but no digit has the value of ten. If a digit having this value were added, the base would cease to be ten and become eleven.
 
Integer: a positive or negative number with no fractional or decimal part. 1, 37 and 111 are all integers, as are -1, -37 and -111.

Magic constant: See magic square.

Magic square: a square of numbers in which rows, columns and diagonals all add up to the same number, which is known as the magic constant.

Reciprocal: the reciprocal of a number is one divided by that number. The reciprocal of 7 is 1/7 (and the reciprocal of 1/7 is 1/1/7 or 7).

Rational number: a number that can be expressed as a ratio of two integers. 0·142857142857..., although it never ends, can be expressed exactly as the ratio of the whole numbers 1 and 7, or 1/7. Pi, which also never ends, cannot be expressed as a ratio of two whole numbers, and is therefore an irrational number. Pi can, however, be approximated by a rational number.

Repunit: a number like 11, 111, or 1,111 that consists of repeated units, or 1’s. A mathematical shorthand for a repunit is a capital “R” subscripted with the number of repeated units. 11 is therefore R2, 111 R3. 11 is the first repunit and 111 the second, however, because although 1 is R1, 1, for obvious reasons, is not a repunit.¹⁴

Triangular numbers: If x dots can be arranged in the form of an equilateral triangle, x is called a triangular number. As will be seen from the diagram, the nth triangular number is found by adding n to all the integers less than n — a quicker way is to use the formula 1/2n(n+1).¹⁵

                            *
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          *      * *      * * *
    *    * *    * * *    * * * *
*  * *  * * *  * * * *  * * * * *

Triplet: a three-digit number like 111, 222, or 666 in which all three digits are the same. 999 is the largest triplet in base ten, but not in higher bases: in base eleven, for example, the largest triplet would be (say) AAA (= 1330 in base ten).

NOTES

1. Its values in these bases are: 111₂ = 7, 111₃ = 13, 111₅ = 31, 111₆ = 43, 111₈ = 73, 111₁₂ = 157, 111₁₄ = 211, 111₁₅ = 241, 111₁₇ = 307, 111₂₀ = 421, 111₂₁ = 463, 111₂₄ = 601, 111₂₇ = 757, 111₃₃  = 1123, 111₃₈ = 1483, 111₄₁ = 1723, 111₅₀ = 2551, 111₅₄ = 2971, 111₅₇ = 3307, 111₅₉ = 3541, 111₆₂ = 3907, 111₆₆ = 4423, 111₆₉ = 4831, 111₇₁ = 5113, 111₇₅ = 5701, 111₇₇ = 6007, 111₇₈ = 6163, 111₈₀ = 6481, 111₈₉ = 8011, 111₉₀ = 8191, 111₉₉ = 9901, 111₁₁₁ = 12433.

2. The Penguin Dictionary of Curious and Interesting Numbers, David Wells, Penguin, London, 1987, entry for “111”, pg. 134

3. This transliteration into Hebrew is not strictly correct, however: q should be k. See Robert Graves’ The White Goddess, ch. 19, “The Number of the Beast”.

4. It’s not in fact surprising that the Hebrew, Greek, and Latin Bibles should all use the same chapter and verse, because they were all completed before the system of using chapters and verses was devised by the Catholic church, which naturally used the same numbering for each.

5. See my articles “Golden Laughter”, in Intense Device (1997), and “Guts’n’Roses”, Headpress JOE #15.

6. The Cambridge Encyclopedia, ed. David Crystal et al., Cambridge University Press, 1990, entry for “Radon”.

7. The average value of any decimal place times the period, or the number of digits before looping begins.

8. Beside magic squares, there are also magic polygrams. The magic hexagram has a magic total of 26 along each “side”. Similarly, a magic heptagram has a magic total of 30, and magic octogram a magic toal of 34. This means a magic pentacontapentagram (with 55 points) should have a magic total of 222. Does it? I don’t know yet: I’m still working on it... (Not any more: see above for the 55-point star with a magic total of 222 and the 166-point star with a magic total of 666).

9. According to the listener’s notes in The Blues Collection 3: Chuck Berry (Orbis Publishing, London, 1993), that is. But when I play the accompanying tape, it isn’t 2’22”.

10. See my article “Guts’n’Roses” in Headpress JOE #15.

11. The golden ratio is (√5 + 1) / 2 or 1·6180339887...; 360 / 222 = 1·621621621..., while 360 / 221 = 1·628959276... and 360 / 223 = 1.61439775...

12. This equals 1 — the probability of no triplets — the probabilities of exactly one, two, three, and four triplets. The probabilities are based on the assumption that 333 separate triplets can occur in the first 1000 digits of phi.

13. The Penguin Dictionary of Curious and Interesting Numbers, entry for “2·718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699...”, pg. 46

14. Ibid., entries for “111”, pg. 134, and “1,111,111,111,111,111,111”, pg. 197

15. Ibid., entry for “15”.