Additional properties of the number 42 include:
- It is the third primary pseudoperfect number.
- It is a Catalan number. Consequently, 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered binary trees with six leaves, the number of ways in which five pairs of nested parentheses can be arranged, etc.
- It is an alternating sign matrix number, that is, the number of 4-by-4 alternating sign matrices.
- It is the number of partitions of 10—the number of ways of expressing 10 as a sum of positive integers (note a different sense of partition from that above).
- Given 27 same-size cubes whose nominal values progress from 1 to 27, a 3 × 3 × 3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of 3 numbers whose sum of values is 42.
- It is the third pentadecagonal number. It is a meandric number and an open meandric number.
- 42 is the only known value that is the number of sets of four distinct positive integers a, b, c, d, each less than the value itself, such that ab − cd, ac − bd, and ad − bc are each multiples of the value. Whether there are other values remains an open question.
- 42 is a (2,6)-perfect number (super-multiperfect), as σ2(n) = σ(σ(n)) = 6n.
- 42 is the resulting number of the original Smith number (4937775 = 3 × 5 × 5 × 65837): Both the sum of its digits (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)) result in 42.
- The dimension of the Borel subalgebra in the exceptional Lie algebra e6 is 42.
- 42 is the largest number n such that there exist positive integers p, q, r with 1 = 1/n + 1/p + 1/q + 1/r
- 42 is the smallest number k such that for every Riemann surface C, #Aut(C) ≤ k deg(KC) = k(2g − 2) (Hurwitz’s automorphisms theorem)
- 42 is a perfect score on the USA Math Olympiad (USAMO) and International Mathematical Olympiad (IMO).
- 42 is the maximum of core points awarded in International Baccalaureate Diploma Programme.
- 42 is the sum of the first 6 positive even numbers.
- 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. 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. (The same idea was proposed, without calculation by Lewis Carroll in 1893 in Sylvie and Bruno Concluded.)
The Hitchhiker’s Guide to the Galaxy
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” was 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 fact that Adams named the episodes of the radio play “fits”, the same archaic title for a chapter or section used by Lewis Carroll in “The Hunting of the Snark”, suggests that Adams was influenced by Carroll’s fascination with and frequent use of the number. 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.
The 2011 book 42: Douglas Adams’ Amazingly Accurate Answer to Life, the Universe and Everything examines Adams’ choice of the number 42 and also contains a compendium of some instances of the number in science, popular culture, and humor.