# ترتيبات المقدار (أعداد)

(تم التحويل من Orders of magnitude (numbers))

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## أصغر من 10−30

• حوسبة: رقم 5×10−324 is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value.
• حوسبة: رقم 1.4×10−45 is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.

## 10−30

(0.000000000000000000000000000001; 1000−10; المقياس القصير: one nonillionth, المقياس الطويل: one quintillionth)

• الرياضيات: The probability in a game of bridge of all four players getting a single-suit hand is approximately 4.47×10−28

## 10−27

(0.000000000000000000000000001; 1000−9; المقياص القصير: one octillionth, المقياص الطويل: one quadrilliardth)

## 10−24

(0.000000000000000000000001; 1000−8; المقياس القصير: one septillionth المقياس الطويل: one quadrillionth)

ISO: يوكتو- (y)

## 10−21

(0.000000000000000000001; 1000−7; المقياص القصير: one sextillionth, المقياص الطويل: one trilliardth)

ISO: زپتو- (z)

• الرياضيات: The probability of matching 20 numbers for 20 in a game of Keno is approximately 2.83 × 10−19.

## 10−18

(0.000000000000000001; 1000−6; المقياص القصير: one quintillionth, المقياص الطويل: one trillionth)

ISO: أتو- (a)

• الرياضيات: The probability of rolling snake eyes 10 times in a row on a pair of fair dice is about 2.74×10−16

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## 10−15

(0.000000000000001; 1000−5; المقياس القصير: one quadrillionth, المقياس الطويل: one billiardth)

ISO: فمتو- (f)

## 10−12

(0.000000000001; 1000−4; المقياس القصير: one trillionth, المقياس الطويل: one billionth)

ISO: پيكو- (p)

• الرياضيات: The probability in a game of bridge of one player getting a single-suit hand is approximately 2.52×10−11 (0.00000000252%)
• BioMed: Human visual sensitivity to 1000 nm light is approximately 1.0×10−10 of its peak sensitivity at 555 nm. 

## 10−9

(0.000000001; 1000−3; المقياس القصير: one billionth; المقياس الطويل: one milliardth)

ISO: نانو- (n)

• Mathematics — Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the US Powerball lottery, with a single ticket, under the rules اعتبارا من أغسطس 2009, are 195,249,053 to 1 against, for a probability of 5.12×10−9 (0.000000512%).
• Mathematics — Lottery: The odds of winning the Jackpot (matching the 6 main numbers) in the UK National Lottery, with a single ticket, under the rules اعتبارا من أغسطس 2009, are 13,983,815 to 1 against, for a probability of 7.15×10−8 (0.00000715%).

## 10−6

(0.000001; 1000−2; المقياسان الطويل والقصير: one millionth)

ISO: ميكرو- (μ)

• Mathematics - Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5 × 10−6 (0.00015%).
• Mathematics — Poker: The odds of being dealt a straight flush (other than a royal flush) in poker are 72,192 to 1 against, for a probability of 1.4 × 10−5 (0.0014%).
• Mathematics — Poker: The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4 × 10−4 (0.024%).

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## 10−3

(0.001; 1000−1; one thousandth)

ISO: ميلي- (m)

• Mathematics — Poker: The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10−3 (0.14%).
• Mathematics — Poker: The odds of being dealt a flush in poker are 507.8 to 1 against, for a probability of 1.9 × 10−3 (0.19%).
• Mathematics — Poker: The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10−3 (0.39%).
• Physics: α = 0.007297352570(5), the fine-structure constant.

## 10−2

(0.01; one hundredth)

ISO: سنتي- (c)

• Mathematics — Lottery: The odds of winning any prize in the UK National Lottery, with a single ticket, under the rules as of 2003, are 54 to 1 against, for a probability of about 0.018 (1.8%)
• Mathematics — Poker: The odds of being dealt a three of a kind in poker are 46 to 1 against, for a probability of 0.021 (2.1%)
• Mathematics — Lottery: The odds of winning any prize in the US Powerball Multistate Lottery, with a single ticket, under the rules as of 2006, are 36.61 to 1 against, for a probability of 0.027 (2.7%)
• Mathematics — Poker: The odds of being dealt two pair in poker are 20 to 1 against, for a probability of 0.048 (4.8%).

## 10−1

(0.1; one tenth)

ISO: ديسي- (d)

• Mathematics — Poker: The odds of being dealt only one pair in poker are about 5 to 2 against (2.37 to 1), for a probability of 0.42 (42%).
• Mathematics — Poker: The odds of being dealt no pair in poker are nearly 1 to 2, for a probability of about 0.5 (50%)
• Legal history: 10% was a widespread as the tax raised for income or produce in the ancient and medieval period, see tithe.

(1; واحد)

(10; ten)

ISO: deca- (da)

## 102

(100; مائة)

ISO: hecto- (h)

• حوسبة: There are 128 characters in the ASCII character set.
• علم الصوتيات: The Taa language is estimated to have between 130 and 164 distinct phonemes
• العلوم الصسياسية: 129 عضو في الأمم المتحدة من 2006.
• التاريخ الاوروبي: groupings of a hundred homesteads was a common administrative unit in Northern Europe and Great Britain, see Hundred (county subdivision)

## 103

(1000; ألف)

ISO: kilo- (k)

• Typesetting: 2,000–3,000 letters on a typical typed page of text.
• التاريخ العسكري: 4,200 (جمهورية) أو5,200 (إمبراطورية) في نظاق Roman legion
• BioMed: the DNA of the simplest viruses has some 5,000 base pairs.
• Linguistic diversity: Estimates for living human languages or dialects range between 5,000 and 10,000 (SIL Ethnologue in 2007 listed 6,912 known living languages).
• Lexicography: 8,674 unique words in the Hebrew Bible

## 104

• BioMed: Each neuron in the human brain is estimated to connect to 10,000 others
• Lexicography: 14,500 unique English words occur in the King James Version of the Bible
• Language: There are 20,000–40,000 distinct Chinese characters, depending on how one counts them
• BioMed: Each human being is estimated to have 30,000 to 40,000 genes
• Records: As of July 2004, the largest number of decimal places of π that have been recited from memory - > 42000
• Mathematics: 65,537 is the largest known Fermat prime

## 105

(100000; مائة ألف أو lakh)

## 106

(1000000; 10002; المقياسان الطويل والقصير: مليون)

ISO: مـِگا- (M)

• BioMed — Species: The World Resources Institute claims that approximately 1.4 million species have been named, out of an unknown number of total species (estimates range between 2 and 100 million species)
• Info: The FreeDB database has around 1 750 000 entries as of June 2005
• Mathematics — Playing cards: There are 2 598 960 different 5-card poker hands that can be dealt from a standard 52-card deck.
• Info — Web sites: as of 26 February 2010, Wikipedia contains approximately 3,206,530 articles in the اللغة الإنگليزية
• Geography/Computing — Geographic places: The NIMA GEOnet Names Server contains approximately 3.88 million named geographical features outside the United States, with 5.34 million names. The USGS Geographic Names Information System claims to have almost 2 million physical and cultural geographic features within the United States
• Genocide: Approximately 5,100,000-6,200,000 Jews were killed in the Holocaust.

## 107

(10000000; المقياسان الطويل والقصير: عشرة مليون)

## 108

(100000000; المقياسان الطويل والقصير: مائة مليون)

## 109

(1000000000; 10003; المقياس القصير: بليون; المقياس الطويل: ألف مليون أو مليار)

ISO: گيگا- (G)

• ديموغرافيا — الهند: 1,096,000,000 - تقريبا عدد سكان الهند في 2007
• ديموغرافيا — الصين: 1,311,000,000 - تقريبا عدد سكان الصين في 2007.
• Computing — Computational limit of a 32-bit CPU: 2 147 483 647 is equal to 231−1, and as such is the largest number which can fit into a signed (two's complement) 32-bit integer on a computer, thus marking the upper computational limit of a 32-bit CPU such as Intel's Pentium-class computer chips.
• BioMed — base pairs in the genome: approximately 3×109 base pairs in the human genome
• حوسبة- IPv4: 4,294,967,296 (232) possible unique IP addresses.
• حوسبة: 4,294,967,296 - the number of bytes in 4 gibibytes; in computation, the 32-bit computers can directly access 232 pieces of address space, this leads directly to the 4 gigabyte limit on main memory.
• رياضيات: 4,294,967,297 is a Fermat number and semiprime. It is the smallest number of the form $2^{2^{n}}+1$ which is not a prime number.
• Demographics — world population: 6,854,035,840 - Estimated total mid-year population for the world in 2010 (July 6).

## 1010

(10000000000; [[المقياسان الطويل والقصير|المقياس القصير]: عشرة بليون; المقياس الطويل: عشرة آلاف مليون أو عشرة مليار)

• BioMed — bacteria in the human body: there are roughly 1010 bacteria in the human oral cavity 
• Astronomy — Observable galaxies: as of 2003 there are between 1×1010 and 8×1010 galaxies in the observable Universe
• Computing — web pages: approximately 5.6×1010 web pages indexed by Google as of 2010.
• 87,178,291,200 = 14!

## 1011

(100000000000; المقياس الطويل: مائة بليون; المقياس الطويل: مائة ألف مليون, أو مائة مليار)

## 1012

(1000000000000; 10004; المقياس القصير: واحد تريليون; المقياس الطويل: واحد بليون)

ISO: تـِترا- (T)

• BioMed — Bacteria on the human body: the surface of the human body houses roughly 1012 bacteria.
• 1,307,674,368,000 = 15!
• علم الأحياء البحرية: 3,500,000,000,000 (3.5 × 1012) - estimated population of fish in the ocean.
• Mathematics — Known digits of π: اعتبارا من 2010, the number of known digits of π is 5,000,000,000,000 (5.0000×1012). 
• Mathematics: 7,625,597,484,987 - a number that often appears when dealing with powers of 3. It can be expressed as $19683^{3}$ , $27^{9}$ , $3^{27}$ , $3^{3^{3}}$ and 33 or when using Knuth's up-arrow notation it can be expressed as $3\uparrow \uparrow 3$ and $3\uparrow \uparrow \uparrow 2$ .
• Mathematics: 1013 - The approximate number of known non-trivial zeros of the Riemann zeta function اعتبارا من 2004.
• BioMed — Cells in the human body: the human body consists of roughly 1014 cells, of which only 1013 are human. The remainder of the cells are bacteria, which mostly reside in the gastrointestinal tract, although the skin is also covered in bacteria.
• BioMed-Insects: 200,000,000,000,000 (2 × 1014) - The estimated number of ants on Earth.
• Computing - MAC-48: 281,474,976,710,656 (248) possible unique physical addresses.
• Mathematics: 953,467,954,114,363 is the largest known Motzkin prime.
• 20,922,789,888,000 = 16!
• 355,687,428,096,000 = 17!

## 1015

(1000000000000000; 10005; المقياس القصير: واحد كوادريليون; المقياس الطويل: ألف بليون أو بليار)

ISO: پـِتا- (P)

• 6,402,373,705,728,000 = 18!
• الرياضيات: 48,988,659,276,962,496 is the fifth taxicab number.
• علم التعمية: There are 7.205759×1016 different possible keys in the obsolete 56 bit DES symmetric cipher.
• 121,645,100,408,832,000 = 19!

## 1018

(1000000000000000000; 10006; المقياس القصير: كوينتيليون; المقياس الطويل: واحد تريليون)

ISO: اكسا- (E)

• 2,432,902,008,176,640,000 = 20!
• Computing — Manufacturing: An estimated 6×1018 transistors were produced worldwide in 2008.
• Computing — Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22×1018) is equal to 263-1, and as such is the largest number which can fit into a signed (two's complement) 64-bit integer on a computer.
• Mathematics — NCAA Basketball Tournament: There are 9,223,372,036,854,775,808 (263) possible ways to enter the bracket.
• BioMed — Insects: It has been estimated that the insect population of the Earth is about 1019.
• Mathematics — Rubik's Cube: There are 43,252,003,274,489,856,000 (about 43×1018) different positions of a 3x3x3 Rubik's Cube
• 51,090,942,171,709,440,000 = 21!
• Password strength: Usage of the 95-character set found on standard computer keyboards for a 10-character password yields a computationally intractable 59,873,693,923,837,890,625 (9510, approximately 6×1019) permutations.
• Economics: Hyperinflation in Zimbabwe estimated in February 2009 by some economists at 10 sextillion percent, or a factor of 1020

## 1021

(1000000000000000000000; 10007; المقياس القصير: واحد سكستيليون; المقياس الطويل: ألف تريليون أو ألف تليار

ISO: زيتا- (Z)

• 1,124,000,727,777,607,680,000 = 22!
• Geo — Grains of sand: all the world's beaches put together have been estimated to hold roughly 1021 grains of sand.
• Mathematics — Sudoku: There are 6,670,903,752,021,072,936,960 (≈6.7×1021) 9×9 sudoku grids.
• 25,852,016,738,884,976,640,000 = 23!
• Astronomy — Stars: 70 sextillion = 7×1022 estimated number of stars within range of telescopes (as of 2003), see mass of the observable universe.
• Mathematics: 146,361,946,186,458,562,560,000 (≈1.5×1023) is the fifth unitary perfect number.
• Chemistry: there are roughly 6.022×1023 molecules in one mole of any substance (Avogadro's number)
• 620,448,401,733,239,439,360,000 = 24!

## 1024

(1000000000000000000000000; 10008; المقياس القصير: واحد سپتيليون; المقياس الطويل: أو كوادريليون)

ISO: يوتا- (Y)

• الرياضيات: 2,833,419,889,721,787,128,217,599 (≈2.8×1024) is a Woodall prime.
• 15,511,210,043,330,985,984,000,000 = 25!
• 403,291,461,126,605,635,584,000,000 = 26!

## 1027

(1000000000000000000000000000; 10009; المقياس القصير: واحد أوكتيليون; المقياس القصير: أو ألف كوادريليون)

• 10,888,869,450,418,352,160,768,000,000 = 27!
• BioMed — Atoms in the human body: the average human body contains roughly 7×1027 atoms, see 
• Mathematics — Poker: the number of unique combinations of hands and shared cards in a 10-player game of Texas Hold'em is approximately 2.117×1028, see Poker probability (Texas hold 'em).
• 304,888,344,611,713,860,501,504,000,000 = 28!

## 1030

(1000000000000000000000000000000; 100010; المقياس القصير: واحد نونليون; المقياس الطويل: واحد كوينتليون)

• BioMed: عدد الخلايا البكتيرية على الأرض
• 8,841,761,993,739,701,954,543,616,000,000 = 29!
• الرياضيات: The partition of 1000 is 24,061,467,864,032,622,473,692,149,727,991.
• 265,252,859,812,191,058,636,308,480,000,000 = 30!
• الرياضيات: 2108 = 324,518,553,658,426,726,783,156,020,576,256 is the last known power of two not containing a digit '9'.

## 1033

(1000000000000000000000000000000000; 100011; المقياس القصير: واحد ديشليون; المقياس القصير: ألف كوينتليون أو كوينتليار)

• الرياضيات: 1,298,074,214,633,706,835,075,030,044,377,087 (≈1.3×1033) is a Carol prime
• Mathematics — Alexander's Star: There are 72,431,714,252,715,638,411,621,302,272,000,000 (about 7.24×1034) different positions of Alexander's Star

## 1036

(1000000000000000000000000000000000000; 100012; المقياس القصير: واحد أونديشيليون; المقياس الطويل: واحد سكستيليون)

• الفيزياء: kee2 / Gm2, the ratio of the electrical to the gravitational forces between two protons, is roughly 1036.
• الحوسبة: The address range of IPv6 (2128) is approximately equal to 3.40282367×1038, and is the theoretical maximum number of Internet addresses that can be allocated under the IPv6 addressing system, one more than the largest value that can be represented by a single-precision IEEE floating-point value, the total number of different Universally Unique Identifiers (UUIDs) that can be generated, and in Cryptography, the total number of different possible keys in the AES 128 bit keyspace (symmetric cipher).

## 1039

(1000000000000000000000000000000000000000; 100013; المقياس القصير: one duodecillion; المقياس الطويل: one thousand sextillion, or one sextilliard)

## 1042 to 10100

(1000000000000000000000000000000000000000000; 100014; المقياس القصير: one tredecillion; المقياس الطويل: واحد سپتيليون)

• الرياضيات: 141×2141+1 = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833 (≈3.9×1044) is the second Cullen prime
• الرياضيات: There are 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (≈7.401×1045) possible permutations for the Rubik's Revenge (4x4x4 Rubik's Cube).
• الشطرنج: 1×1050 is an estimate of the number of legal chess positions.
• الرياضيات: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (≈8×1053) is order of Monster group
• علم التعمية: There are 6.27710174×1057 different possible keys in the AES 192 bit keyspace (symmetric cipher).
• Cosmology: 8×1060 is roughly the number of Planck time intervals since the universe is theorised to have been created in the Big Bang 13.7 ± 0.2 billion years ago.
• Cosmology: 1×1063 is Archimedes’ estimate in The Sand Reckoner of the total number of grains of sand that could fit into the entire cosmos, the diameter of which he estimated to be what we call 2 light years.
• Mathematics — Cards: 52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 (≈8×1067) - the number of ways to order the cards in a 52-card deck.
• Mathematics: 1,808,422,353,177,349,564,546,512,035,512,530,001,279,481,259,854,248,860,454,348,989,451,026,887 (≈1.8×1072) - The largest known prime factor found by ECM factorization اعتبارا من 2010.
• الرياضيات: There are 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 (2.8287×1074) possible permutations for the Professor's Cube (5x5x5 Rubik's Cube).
• علم التعمية: There are 1.15792089×1077 different possible keys in the AES 256 bit keyspace (symmetric cipher).
• Cosmology: various sources estimate the total number of fundamental particles in the observable universe in the range 1080 to 1085. However, these estimates are generally regarded as guesswork.
• Mathematics: 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000; 10100, a googol

## Larger than 10100

• Mathematics: There are 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000 000 000 000 000 000 000 (1.5715 ×10116) distinguishable permutations of the V-Cube 6 (6x6x6 Rubik's Cube).
• Chess: عدد شانون, 10120, an estimation of the game-tree complexity of chess.
• الفيزياء: 8×10120, ratio of the mass-energy in the observable universe to the energy of a photon with a wavelength the size of the observable universe.
• تاريخ — الرياضيات: Asankhyeya is equal to 10140 in ancient India.
• Xiangqi: 10150, an estimation of the game-tree complexity of xiangqi.
• 100! = approx. 9,33262154 * 10157
• الرياضيات: There are 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 (1.9501 ×10160) distinguishable permutations of the V-Cube 7 (7x7x7 Rubik's Cube).
• Board games: 3.457×10181, number of ways to arrange the tiles in English Scrabble on a standard 15-by-15 Scrabble board.
• الفيزياء: 4×10185, approximate number of Planck volumes in the observable universe.
• الحوسبة: 1.797 693 134 862 3157×10308 is approximately equal to the largest value that can be represented by a double-precision IEEE floating-point number.
• Go: 10365, an estimation of the game-tree complexity in the game of Go.[بحاجة لمصدر]
• 200! = approx. 7,886579*10374
• الرياضيات: 26384405 + 44052638 is a 15,071-digit Leyland prime; the largest which has been proven اعتبارا من 2010.
• 10,000! = 2,84626*1035659
• الرياضيات: 183,027 × 2265,440−1 is a 79,911-digit Sophie Germain prime; the largest known اعتبارا من مارس 2010.
• الرياضيات: 65,516,468,355 · 2333,333 ± 1 are 100,355-digit twin primes; the largest known اعتبارا من 2010.
• الرياضيات: 392,113# + 1 is a 169,966-digit primorial prime; the largest known اعتبارا من 2010.
• الرياضيات: 10200000 + 47960506974×1099995 + 1 is a 200,001-digit palindromic prime, the largest known اعتبارا من أكتوبر 2010.
• الرياضيات: approximately 7.76 · 10206,544 cattle in the smallest herd which satisfies the conditions of the Archimedes' cattle problem.
• الرياضيات: 94,550! – 1 is a 429,390-digit factorial prime; the largest known اعتبارا من 2010.
• الرياضيات: 24,518262,144 + 1 is a 1,150,678-digit [[Generalized Fermat prime]], the largest known اعتبارا من 2010.
• Mathematics: 19,249 × 213,018,586 + 1 is a 3,918,990-digit Proth prime, the largest known اعتبارا من 2010.
• الرياضيات: 243,112,609 − 1 is a 12,978,189-digit Mersenne prime; the largest known prime of any kind اعتبارا من 2010.
• الرياضيات: 243,112,608 × (243,112,609 − 1) is a 25,956,377-digit perfect number, the largest known as of 2010.
• الرياضيات — History: 1080,000,000,000,000,000, largest named number in Archimedes' Sand Reckoner.
• الرياضيات: 10googol ($10^{10^{100}}$ ), a googolplex.
• الرياضيات: $10^{\,\!10^{10^{34}}}$ , order of magnitude of an upper bound that occurred in a proof of Skewes (later estimated to actually be near 1.397 × 10316).
• الرياضيات: $10^{\,\!10^{10^{1000}}}$ , order of magnitude of another upper bound in a proof of Skewes.
• الرياضيات: Moser's number "2 in a mega-gon" is approximately equal to 10↑↑↑...↑↑↑10, where there are 10↑↑257 arrows.
• الرياضيات: Graham's number, the last ten digits of which are ...24641 95387. Arises as an upper bound solution to a problem in Ramsey theory and is probably the largest number seriously used in a mathematical proof. Representation in powers of 10 would be impractical (the number of digits in the exponent far exceeds the number of particles in the observable universe).

## المصادر

1. ^ Bridge hands
2. ^ P. L. Walraven and H. J. Lebeek. "Foveal Sensitivity of the Human Eye in the Near Infrared." J. Opt. Soc. Am. 53, 765-766 (1963). http://www.opticsinfobase.org/abstract.cfm?URI=josa-53-6-765
3. ^ The On-Line Encyclopedia of Integer Sequences
4. ^ Earth microbes on the Moon
5. ^ "Earth microbes on the moon". Science@Nasa. Retrieved 7 July 2008.
6. ^ 5 Trillion Digits of Pi - New World Record By Alexander J. Yee & Shigeru Kondo
7. ^ http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html
8. ^ Savage, D. C. (1977). "Microbial Ecology of the Gastrointestinal Tract". Annual Review of Microbiology. 31: 107. doi:10.1146/annurev.mi.31.100177.000543. PMID 334036.
9. ^ Berg, R. (1996). "The indigenous gastrointestinal microflora". Trends in Microbiology. 4: 430. doi:10.1016/0966-842X(96)10057-3.
10. ^ http://www.sia-online.org/cs/papers_publications/press_release_detail?pressrelease.id=96
11. ^
12. ^ "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. Retrieved 14 March 2009.
13. ^ To see the Universe in a Grain of Taranaki Sand
14. ^ Sudoku enumeration
15. ^ http://msowww.anu.edu.au/news/media_releases/media_release_030717.php
16. ^ How many atoms are in the human body?
17. ^ (sequence A070177 in OEIS)
18. ^ (sequence A035064 in OEIS)
19. ^ Victor Allis (1994). Searching for Solutions in Games and Artificial Intelligence (PDF). Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 9090074880. Retrieved 2010-02-09.
20. ^