What does the number 42 mean in The Hitchhiker's Guide to the Galaxy?

Natural number

42 (forty-two) is the natural number that follows 41 and precedes 43.

Mathematics

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Forty-two (42) is the sixth pronic number[1] and the eighth abundant number,[2] with an abundance of 12,[3] equal to the average of its eight divisors as an arithmetic number.[4][5]

Its prime factorization 2 × 3 × 7 {\displaystyle 2\times 3\times 7} makes it the second sphenic number, and also the second of the form (2.3.r).[6] 42 is the aliquot sum of 30,[7] the smallest sphenic number and second number to have an abundance of 12 after 24, and preceding 42. 42 itself has an aliquot sum of 54; within an aliquot sequence of twelve composite numbers (42,54,66,78,90,144,259,45,33,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.

It is also the sum of the first six positive non-zero even numbers, 2 + 4 + 6 + 8 + 10 + 12 {\displaystyle 2+4+6+8+10+12} , and a Harshad number in decimal, because the sum of its digits is six ( 4 + 2 = 6 ) {\displaystyle (4+2=6)} , which evenly divides 42.[8]

42 is the fifth Catalan number, following 14; consequently, it is[9]

• the number of triangulations of a heptagon.
• the number of rooted ordered binary trees with six leaves, and
• the number of noncrossing partitions of a set of five elements, as well as the ways in which five pairs of nested parentheses can be arranged, etc.

Additionally, 42 is the smallest number k {\displaystyle k} that is equal to the sum its non-prime proper divisors; i.e. 42 = 1 + 6 + 14 + 21 {\displaystyle 42=1+6+14+21} [10] (with the latter term representing the sixth triangular number).[11]

42 is also the third primary pseudoperfect number,[12] and the first (2,6)-perfect number (super-multiperfect), where σ 2 ( n ) = σ ( σ ( n ) ) = 6 n . {\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=6n.} [13]

42 is the number of integer partition of 10: the number of ways of expressing 10 as a sum of positive integers.[14] 1111123, one of the forty-two unordered integer partitions of 10, has 42 ordered compositions, since 7 ! ÷ 5 ! = 42. {\displaystyle 7!\div 5!=42.}

As a polygonal number, 42 is the first (non-trivial) fifteen-sided pentadecagonal number.[15] It is also the fourth meandric number,[16] and seventh open meandric number[17] (following 8 and 14, respectively).

On the other hand, an angle of 42 degrees can be constructed with a compass and straight edge with the use of the golden ratio; i.e. through the difference between constructible angles of 60 and 18 degrees (with root pentagonal symmetry).

Where the plane-vertex tiling 3.10.15 is constructible through elementary methods, the largest such tiling, 3.7.42, is not. This means that the 42-sided tetracontadigon is the largest such regular polygon that can only tile a vertex alongside other regular polygons, without tiling the plane.[18][19][20]

42 is also the first non-trivial hendecagonal (11-gonal) pyramidal number, after 12.[21][22][a] Otherwise, forty-two is the least possible number of diagonals of a simple convex hendecahedron (or 11-faced polyhedron).[28][29][b]

42 is the only known k {\displaystyle k} that is equal to the number of sets of four distinct positive integers ( a , b , c , d ) {\displaystyle (a,b,c,d)} — each less than k {\displaystyle k} — such that   a b − c d ,   {\displaystyle {\text{ }}ab-cd,{\text{ }}} a c − b d   {\displaystyle ac-bd{\text{ }}} and   a d − b c   {\displaystyle {\text{ }}ad-bc{\text{ }}} are all multiples of k {\displaystyle k} . Whether there are other values remains an open question.[30]

42 is the resulting number of the original Smith number: 4937775 = 3 × 5 × 5 × 65837. {\displaystyle 4937775=3\times 5\times 5\times 65837.} Both the sum of its digits, 4 + 9 + 3 + 7 + 7 + 7 + 5 {\displaystyle 4+9+3+7+7+7+5} , and the sum of the digits in its prime factorization, 3 + 5 + 5 + ( 6 + 5 + 8 + 3 + 7 ) {\displaystyle 3+5+5+(6+5+8+3+7)} , result in 42.[31]

42 is the number of isomorphism classes of all simple and oriented directed graphs on four vertices.[32] I.e., the number of all outcomes (up to isomorphism) of a tournament of four teams where a game between a pair of teams results in three possible outcomes: wins from either team, or a draw.[33]

42 is the fourth Robbins number, equivalently the number of 4 × 4 {\displaystyle 4\times 4} alternating sign matrices.[34][35] It is also the number of ways to arrange the numbers 1 {\displaystyle 1} through 9 {\displaystyle 9} in a 3 × 3 {\displaystyle 3\times 3} matrix such that the numbers in each row and column are in ascending order.

The 3 × 3 × 3 simple magic cube with rows summing to 42

42 is the magic constant of the smallest non-trivial magic cube, a 3 × 3 × 3 {\displaystyle 3\times 3\times 3} cube with entries of 1 through 27, where every row, column, corridor, and diagonal passing through the center sums to forty-two.[36][37]

42 is the number of (3, 3, 3) standard Young tableaux that use distinct entries[38][39][40] (as well as the number of (2, 2, 2, 2, 2) tableaux).[41][42]

The last natural number less than 100 whose representation as a sum of three cubes was found (in 2019) is forty-two, where,[43]

80 , 435 , 758 , 145 , 817 , 515 3 + 12 , 602 , 123 , 297 , 335 , 631 3 + ( − 80 , 538 , 738 , 812 , 075 , 974 ) 3 = 42. {\displaystyle 80,435,758,145,817,515^{3}+12,602,123,297,335,631^{3}+(-80,538,738,812,075,974)^{3}=42.}

The dimension of the Borel subalgebra in the exceptional Lie algebra e6 is 42.

42 is the smallest number k {\displaystyle k} such that for every Riemann surface C {\displaystyle \mathbf {C} } of genus g ≥ 2 {\displaystyle g\geq 2} , # Aut ( C ) ≤ k   deg ( K C ) = k ( 2 g − 2 ) {\displaystyle \#{\text{Aut}}(C)\leq k\ {\text{deg}}(K_{C})=k(2g-2)} (by the Hurwitz's automorphisms theorem).

This is related to 42 being the largest n {\displaystyle n} where there exist positive integers p , q , r {\displaystyle p,q,r} whose reciprocals alongside that of forty-two generate the sum,[44]

1 = 1 2 + 1 3 + 1 7 + 1 42 . {\displaystyle 1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{42}}.}

Notice that the first three unit fractions are the first values in the infinite series (of Egyptian fractions) that most rapidly converges to 1 {\displaystyle 1} (see, Sylvester's sequence).

Other properties

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42 is the smallest integer that can only be made from a minimal number of fours (seven) using only addition, subtraction, multiplication, and division, where an intermediate value has to be a non-integer:[citation needed]

42 = ( 4 + 4 ) × ( 4 + 4 + 4 4 4 ) = 8 × 5.25. {\displaystyle 42=(4+4)\times (4+{\frac {4+{\frac {4}{4}}}{4}})=8\times 5.25.}

In decimal representation, the first three digits of pi, 3.14 … ≈ π {\displaystyle 3.14\ldots \approx \pi } , can be arranged as a set of two strings to yield: 3 × 14 = 42. {\displaystyle 3\times 14=42.}

In the terminating decimal of the approximation for pi, the string 42 42 42 {\displaystyle 42\;42\;42} occurs at the 242424th decimal "position" (when treating the decimal point as a position, as well).[45]

Science

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• 42 is the atomic number of molybdenum.
• 42 is the atomic mass of one of the naturally occurring stable isotopes of calcium.
• The angle rounded to whole degrees for which a rainbow appears (the critical angle).
• In 1966, mathematician Paul Cooper theorized that the fastest, most efficient way to travel across continents would be to bore a straight hollow tube directly through the Earth, connecting a set of antipodes, remove the air from the tube and fall through.[46] The first half of the journey consists of free-fall acceleration, while the second half consists of an exactly equal deceleration. The time for such a journey works out to be 42 minutes. Even if the tube does not pass through the exact center of the Earth, the time for a journey powered entirely by gravity (known as a gravity train) always works out to be 42 minutes, so long as the tube remains friction-free, as while the force of gravity would be lessened, the distance traveled is reduced at an equal rate.[47][48] (The same idea was proposed, without calculation by Lewis Carroll in 1893 in Sylvie and Bruno Concluded.[49]) Now we know that is inaccurate, and it only would take about 38 minutes.[50]
• As determined by the Babylonians, in 79 years, Mars orbits the Sun almost exactly 42 times.[51]
• The hypothetical efficiency of converting mass to energy, as per

  E = m c 2 , {\displaystyle E=mc^{2},}

  [52]
• In Powers of Ten by Ray and Charles Eames, the known universe from large-scale to small-scale is represented by 42 different powers of ten. These powers range from 1025 meters to 10−17 meters.

Technology

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Astronomy

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Religion

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Popular culture

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The Hitchhiker's Guide to the Galaxy

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The Answer to the Ultimate Question of Life, The Universe, and Everything

The number 42 is, in The Hitchhiker's Guide to the Galaxy by Douglas Adams, the "Answer to the Ultimate Question of Life, the Universe, and Everything", calculated by an enormous supercomputer named Deep Thought over a period of 7.5 million years. Unfortunately, no one knows what the question is. Thus, to calculate the Ultimate Question, a special computer the size of a small planet was built from organic components and named "Earth". The Ultimate Question "What do you get when you multiply six by nine"[66] is found by Arthur Dent and Ford Prefect in the second book of the series, The Restaurant at the End of the Universe. This appeared first in the radio play and later in the novelization of The Hitchhiker's Guide to the Galaxy.

The fourth book in the series, the novel So Long, and Thanks for All the Fish, contains 42 chapters. According to the novel Mostly Harmless, 42 is the street address of Stavromula Beta. In 1994, Adams created the 42 Puzzle, a game based on the number 42. Adams says he picked the number simply as a joke, with no deeper meaning.

Google also has a calculator easter egg when one searches "the answer to the ultimate question of life, the universe, and everything." Once typed (all in lowercase), the calculator answers with the number 42.[67]

In Hervé Le Tellier's novel The Anomaly, a top-secret US Government protocol receives code number 42, inspired by this source.[citation needed]

Works of Lewis Carroll

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Lewis Carroll, who was a mathematician,[68] made repeated use of this number in his writings.[69]

Examples of Carroll's use of 42:

• Alice's Adventures in Wonderland has 42 illustrations.
• Alice's attempts at multiplication (chapter two of Alice in Wonderland) work if one uses base 18 to write the first answer, and increases the base by threes to 21, 24, etc. (the answers working up to 4 × 12 = "19" in base 39), but "breaks" precisely when one attempts the answer to 4 × 13 in base 42, leading Alice to declare "oh dear! I shall never get to twenty at that rate!"[

  citation needed

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• Rule Forty-two in Alice's Adventures in Wonderland ("All persons more than a mile high to leave the court").
• Rule 42 of the Code in the preface[70] to The Hunting of the Snark ("No one shall speak to the Man at the Helm").
• In "fit the first" of The Hunting of the Snark the Baker had "forty-two boxes, all carefully packed, With his name painted clearly on each."[71]
• The White Queen announces her age as "one hundred and one, five months and a day", which—if the best possible date is assumed for the action of Through the Looking-Glass (e.g., a date is chosen such that the rollover from February to March is excluded from what would otherwise be an imprecise measurement of "five months and a day")—gives a total of 37,044 days. If the Red Queen, as part of the same chess set, is regarded as the same age, their combined age is 74,088 days, or 42 × 42 × 42.[72]

Music

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Television and film

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Video games

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Sports

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Jackie Robinson in his now-retired number 42 jersey

Architecture

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• The architects of the Rockefeller Center in New York City worked daily in the Graybar Building where on "the twenty-fifth floor, one enormous drafting room contained forty-two identical drawing boards, each the size of a six-seat dining room table; another room harboured twelve more, and an additional fourteen stood just outside the principals' offices at the top of the circular iron staircase connecting 25 to 26".[77]
• In the Rockefeller Center (New York City) there are a total of "forty-two elevators in five separate banks"[78] which carry tenants and visitors to the sixty-six floors.

Comics

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• Miles Morales was bitten by a spider bearing the number 42, causing him to become a Spider-Man. The number was later heavily referenced in the film Spider-Man: Into the Spider-Verse. The use of 42 within the franchise references Jackie Robinson's use of the number, though many fans incorrectly believed it to be a Hitchhiker's Guide to the Galaxy reference.[79]

Other fields

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Other languages

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Language Translation Afrikaans

twee-en-veertig

Albanian

dyzetedy

Arabic

إثنان و أربعون

(ʾithnān wa ʾarbaʿūn) Armenian

քառասուներկու

(karasunerku) Armenian (Classic)

ԽԲ

(khe ben) Basque

berrogeita bi

Bangla

biyallis ৪২ বিয়াল্লিশ

Belarusian

сорак два

(sorak dva) Bosnian

četrdeset dva

Bulgarian

четиридесет и две

(četirideset i dve) Catalan

quaranta-dos

Chinese

四十二

(

肆拾贰

) (sìshí'èr) Chuvash хĕрĕх иккĕ (xĕrĕx ikkĕ, IIXXXX) Croatian

četrdeset dva

Czech

čtyřicet dva

Danish

toogfyrre

Dhivehi Saalhees Dheyh Dutch

tweeënveertig

Esperanto

kvardek du

Estonian

nelikümmend kaks

Finnish

neljäkymmentäkaksi

Filipino

apatnapu't dalawa

French

quarante-deux

West Frisian

twaenfjirtich

Galician

corenta e dous

Georgian

ორმოცდაორი

(ormocdaori) German

zweiundvierzig

Greek

σαράντα δύο

(saránta dýo) Gujarati betalis Hebrew

ארבעים ושתיים

(arbayim u-shtayim) Hindi

बयालीस, ४२

(bayālīs) Hungarian

negyvenkettő

Icelandic

fjörutíu og tveir

Indonesian

empat puluh dua

Irish

daichead a dó

Italian

quarantadue

Japanese

四十二 (よんじゅうに)

(yonjūni) Kazakh

қырық екі

(qırıq eki) Korean

사십이 / 마흔둘

(sasibi/maheundul) Kannada

ನಲವತ್ತು ಎರಡು

(nalavatthu eradu) Latin

quadraginta duo

Latvian

četrdesmit divi

Livonian

nēļakimdõ kakš

Lithuanian

keturiasdešimt du

Lojban

vore

Luxembourgish

zweeavéierzeg

Macedonian

четириесет и два

(četirieset i dva) Malayalam

നാല്പത്തിരണ്ടു

Maltese

tnejn u erbgħin

Māori

whā tekau ma rua

Marathi bechalis Mongolian

дөчин хоёр

(döchin khoyor) Norwegian

førtito

Pashto

دوه څلوېښت

Persian

چهل و دو

(chehel o du) Polish

czterdzieści dwa

Portuguese

quarenta e dois

Romanian

patruzeci și doi

Russian

сорок два

(sorok dva) Sanskrit

द्विचत्वारिंशत्, ४२

(dvicatvāriṃśat) Serbian

четрдесет два

(četrdeset dva) Shona

Makumi mana nemaviri

Sinhala

හතලිස් දෙක

(hathalis deka) Slovene

dvainštirideset

Slovak

štyridsaťdva

Somali

laba iyo afartan

Spanish

cuarenta y dos

Swedish

fyrtiotvå

Tagalog

apatnapu't dalawa

Tamil

நாற்பத்திரண்டு

(narpatti errundu) Telugu

నలభై రెండు

(nalabai rendu) Thai

สี่สิบสอง

Turkish

kırk iki

Ukrainian

сорок два

(sorok dva) Urdu

بیالیس

(bayālīs) Vietnamese

bốn mươi hai

Volapük

foldegtel

Welsh

pedwar deg dau

/

dau-ar-ddeugain

Yoruba

mejilelogoji

See also

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Notes

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1. ^[11][23] that is also the third sphenic number, following 42 and 30.[6] These first three sphenic numbers are also consecutive (fifth, sixth, and seventh) members in Lemming's simulation sequence, where opposing triangles (starting with just one) are successively joined at vertices (without overlaps in the interior); in this sequence, values represent the total number of triangles joined at each generational step.[24][25] The sum of these three terms

   30 + 42 + 66 = 138

   
   Where 42 is the twenty-eighth [26] the number of twenty-eighth 28-gonal pyramidal number into distinct 28-gonal pyramidal numbers is 42.[27]

   The eleventh triangular number is 66 (and sixth hexagonal number ),that is also the third sphenic number, following 42 and 30.These first three sphenic numbers are also consecutive (fifth, sixth, and seventh) members insequence, where opposing triangles (starting with just one) are successively joined at vertices (without overlaps in the); in this sequence, values represent the total number of triangles joined at each generational step.The sum of these three terms, which is the ninth term.Where 42 is the twenty-eighth composite number the number of integer partitions of the-gonal pyramidal number into distinct 28-gonal pyramidal numbers is 42.

2. ^

   n {\displaystyle n}

   [29][1]

   The sequence of minimum diagonals by such-faced polyhedra follows the sequence of pronic numbers, whose indexes start with 4 (for a square), rather than 0.

References

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Media related to 42 (number) at Wikimedia Commons

A team led by Andrew Sutherland of MIT and Andrew Booker of Bristol University has solved the final piece of a famous 65-year old math puzzle with an answer for the most elusive number of all: 42.

The number 42 is especially significant to fans of science fiction novelist Douglas Adams’ “The Hitchhiker’s Guide to the Galaxy,” because that number is the answer given by a supercomputer to “the Ultimate Question of Life, the Universe, and Everything.”   

Booker also wanted to know the answer to 42. That is, are there three cubes whose sum is 42?

This sum of three cubes puzzle, first set in 1954 at the University of Cambridge and known as the Diophantine Equation x3+y3+z3=k, challenged mathematicians to find solutions for numbers 1-100. With smaller numbers, this type of equation is easier to solve: for example, 29 could be written as 33 + 13 + 13, while 32 is unsolvable. All were eventually solved, or proved unsolvable, using various techniques and supercomputers, except for two numbers: 33 and 42.

Booker devised an ingenious algorithm and spent weeks on his university’s supercomputer when he recently came up with a solution for 33. But when he turned to solve for 42, Booker found that the computing needed was an order of magnitude higher and might be beyond his supercomputer’s capability. Booker says he received many offers of help to find the answer, but instead he turned to his friend Andrew "Drew" Sutherland, a principal research scientist in the Department of Mathematics. “He’s a world’s expert at this sort of thing,” Booker says.

Sutherland, whose specialty includes massively parallel computations, broke the record in 2017 for the largest Compute Engine cluster, with 580,000 cores on Preemptible Virtual Machines, the largest known high-performance computing cluster to run in the public cloud.

Like other computational number theorists who work in arithmetic geometry, he was aware of the “sum of three cubes” problem. And the two had worked together before, helping to build the L-functions and Modular Forms Database (LMFDB), an online atlas of mathematical objects related to what is known as the Langlands Program. “I was thrilled when Andy asked me to join him on this project,” says Sutherland.

Booker and Sutherland discussed the algorithmic strategy to be used in the search for a solution to 42. As Booker found with his solution to 33, they knew they didn’t have to resort to trying all of the possibilities for x, y, and z.

“There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion).

Otherwise, the main difference between the search for 33 and the search for 42 would be the size of the search and the computer platform used. Thanks to a generous offer from UK-based Charity Engine, Booker and Sutherland were able to tap into the computing power from over 400,000 volunteers’ home PCs, all around the world, each of which was assigned a range of values for d. The computation on each PC runs in the background so the owner can still use their PC for other tasks.

Sutherland is also a fan of Douglas Adams, so the project was irresistible.

The method of using Charity Engine is similar to part of the plot surrounding the number 42 in the "Hitchhiker" novel: After Deep Thought’s answer of 42 proves unsatisfying to the scientists, who don’t know the question it is meant to answer, the supercomputer decides to compute the Ultimate Question by building a supercomputer powered by Earth … in other words, employing a worldwide massively parallel computation platform.

“This is another reason I really liked running this computation on Charity Engine — we actually did use a planetary-scale computer to settle a longstanding open question whose answer is 42.”

They ran a number of computations at a lower capacity to test both their code and the Charity Engine network. They then used a number of optimizations and adaptations to make the code better suited for a massively distributed computation, compared to a computation run on a single supercomputer, says Sutherland.

Why couldn't Bristol's supercomputer solve this problem?

“Well, any computer *can* solve the problem, provided you are willing to wait long enough, but with roughly half a million PCs working on the problem in parallel (each with multiple cores), we were able to complete the computation much more quickly than we could have using the Bristol machine (or any of the machines here at MIT),” says Sutherland.  

Using the Charity Engine network is also more energy-efficient. “For the most part, we are using computational resources that would otherwise go to waste,” says Sutherland. “When you're sitting at your computer reading an email or working on a spreadsheet, you are using only a tiny fraction of the CPU resource available, and the Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BOINC), takes advantage of this. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.”

Sutherland and Booker ran the computations over several months, but the final successful run was completed in just a few weeks. When the email from Charity Engine arrived, it provided the first solution to x3+y3+z3=42:

42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

“When I heard the news, it was definitely a fist-pump moment,” says Sutherland. “With these large-scale computations you pour a lot of time and energy into optimizing the implementation, tweaking the parameters, and then testing and retesting the code over weeks and months, never really knowing if all the effort is going to pay off, so it is extremely satisfying when it does.”

Booker and Sutherland say there are 10 more numbers, from 101-1000, left to be solved, with the next number being 114.

But both are more interested in a simpler but computationally more challenging puzzle: whether there are more answers for the sum of three cubes for 3.

“There are four very easy solutions that were known to the mathematician Louis J. Mordell, who famously wrote in 1953, ‘I do not know anything about the integer solutions of x3 + y3 + z3 = 3 beyond the existence of the four triples (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4); and it must be very difficult indeed to find out anything about any other solutions.’ This quote motivated a lot of the interest in the sum of three cubes problem, and the case k=3 in particular. While it is conjectured that there should be infinitely many solutions, despite more than 65 years of searching we know only the easy solutions that were already known to Mordell. It would be very exciting to find another solution for k=3.”