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A Hitchhiker’s Guide to the Number 42

www.scientificamerican.com

Here is how a perfectly ordinary number captured the interest of sci-fi enthusiasts, geeks and mathematicians

Everyone loves unsolved mysteries. Examples include Amelia Earhart’s disappearance over the Pacific in 1937 and the daring escape of inmates Frank Morris and John and Clarence Anglin from Alcatraz Island in California in 1962. Moreover our interest holds even if the mystery is based on a joke. Take author Douglas Adams’s popular 1979 science-fiction novel The Hitchhiker’s Guide to the Galaxy, the first in a series of five. Toward the end of the book, the supercomputer Deep Thought reveals that the answer to the “Great Question” of “Life, the Universe and Everything” is “forty-two.”

Deep Thought takes 7.5 million years to calculate the answer to the ultimate question. The characters tasked with getting that answer are disappointed because it is not very useful. Yet, as the computer points out, the question itself was vaguely formulated. To find the correct statement of the query whose answer is 42, the computer will have to build a new version of itself. That, too, will take time. The new version of the computer is Earth. To find out what happens next, you’ll have to read Adams’s books.

The author’s choice of the number 42 has become a fixture of geek culture. It’s at the origin of a multitude of jokes and winks exchanged between initiates. If, for example, you ask your search engine variations of the question “What is the answer to everything?” it will most likely answer “42.” Try it in French or German. You’ll often get the same answer whether you use Google, Qwant, Wolfram Alpha (which specializes in calculating mathematical problems) or the chat bot Web app Cleverbot.

Since the first such school was created in France in 2013 there has been a proliferation of private computer-training institutions in the “42 Network,” whose name is a clear allusion to Adams’s novels. Today the founding company counts more than 15 campuses in its global network. The number 42 also appears in different forms in the film Spider-Man: Into the Spider-Verse. Many other references and allusions to it can be found, for example, in the Wikipedia entry for “42 (number).”

The number 42 also turns up in a whole string of curious coincidences whose significance is probably not worth the effort to figure out. For example:

In ancient Egyptian mythology, during the judgment of souls, the dead had to declare before 42 judges that they had not committed any of 42 sins.

The marathon distance of 42.195 kilometers corresponds to the legend of how far the ancient Greek messenger Pheidippides traveled between Marathon and Athens to announce victory over the Persians in 490 B.C. (The fact that the kilometer had not yet been defined at that time only makes the connection all the more astonishing.)

Ancient Tibet had 42 rulers. Nyatri Tsenpo, who reigned around 127 B.C., was the first. And Langdarma, who ruled from 836 to 842 A.D. (i.e., the 42nd year of the ninth century), was the last.

The Gutenberg Bible, the first book printed in Europe, has 42 lines of text per column and is also called the “Forty-Two-Line Bible.”

According to a March 6 Economist blog post marking the 42nd anniversary of the radio program The Hitchhiker’s Guide to the Galaxy, which preceded the novel, “the 42nd anniversary of anything is rarely observed.”

A Purely Arbitrary Choice

An obvious question, which indeed has been asked, is whether the use of 42 in Adams’s books had any particular meaning for the author. His answer, posted in the online discussion group alt.fan.douglas-adams, was succinct: “It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one. Binary representations, base thirteen, Tibetan monks are all complete nonsense. I sat at my desk, stared into the garden and thought ’42 will do.’ I typed it out. End of story.”

In the binary system, or base 2, 42 is written as 101010, which is pretty simple and, incidentally, prompted a few fans to hold parties on October 10, 2010 (10/10/10). The reference to base 13 in Adams’s answer requires a more indirect explanation. In one instance, the series suggests that 42 is the answer to the question “What do you get if you multiply six by nine?” That idea seems absurd because 6 x 9 = 54. But in base 13, the number expressed as “42” is equal to (4 x 13) + 2 = 54.

Apart from allusions to 42 deliberately introduced by computer scientists for fun and the inevitable encounters with it that crop up when you poke around a bit in history or the world, you might still wonder whether there is anything special about the number from a strictly mathematical point of view.

Mathematically Unique?

The number 42 has a range of interesting mathematical properties. Here are some of them:

The number is the sum of the first three odd powers of two—that is, 21 + 23 + 25 = 42. It is an element in the sequence a(n), which is the sum of n odd powers of 2 for n > 0. The sequence corresponds to entry A020988 in The On-Line Encyclopedia of Integer Sequences (OEIS), created by mathematician Neil Sloane. In base 2, the nth element may be specified by repeating 10 n times (1010 … 10). The formula for this sequence is a(n) = (2/3)(4n – 1). As n increases, the density of numbers tends toward zero, which means that the numbers belonging to this list, including 42, are exceptionally rare.

The number 42 is the sum of the first two nonzero integer powers of six—that is, 61 + 62 = 42. The sequence b(n), which is the sum of the powers of six, corresponds to entry A105281 in OEIS. It is defined by the formulas b(0) = 0, b(n) = 6b(n – 1) + 6. The density of these numbers also tends toward zero at infinity.

Forty-two is a Catalan number. These numbers are extremely rare, much more so than prime numbers: only 14 of the former are lower than one billion. Catalan numbers were first mentioned, under another name, by Swiss mathematician Leonhard Euler, who wanted to know how many different ways an n-sided convex polygon could be cut into triangles by connecting vertices with line segments. The beginning of the sequence (A000108 in OEIS) is 1, 1, 2, 5, 14, 42, 132…. The nth element of the sequence is given by the formula c(n) = (2n)! / (n!(n + 1)!). And like the two preceding sequences, the density of numbers is null at infinity.

Catalan numbers are named after Franco-Belgian mathematician Eugène Charles Catalan (1814–1894), who discovered that c(n) is the number of ways to arrange n pairs of parentheses according to the usual rules for writing them: a parenthesis is never closed before it has been opened, and one can only close it when all the parentheses that were subsequently opened are themselves closed.

For example, c(3) = 5 because the possible arrangements of three pairs of parentheses are:

( ( ( ) ) ); ( ) ( ) ( ); ( ( ) ) ( ); ( ( ) ( ) ); ( ) ( ( ) )

Forty-two is also a “practical” number, which means that any integer between 1 and 42 is the sum of a subset of its distinct divisors. The first practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66 and 72 (sequence A005153 in OEIS). No simple known formula provides the nth element of this sequence.

All this is amusing, but it would be wrong to say that 42 is really anything special mathematically. The numbers 41 and 43, for example, are also elements of many sequences. You can explore the properties of various numbers on Wikipedia.

What makes a number particularly interesting or uninteresting is a question that mathematician and psychologist Nicolas Gauvrit, computational natural scientist Hector Zenil and I have studied,…

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