P42

• It is the number of isomorphism classes of all simple and oriented directed graphs on 4 vertices. In other words, it is the number of all possible outcomes (up to isomorphism) of a tournament consisting of 4 teams where the game between any pair of teams results in three possible outcomes: the first team wins, the second team wins, or there is a draw. The group stage of the FIFA World cup is a good example.
• It is the third primary pseudoperfect number.^[2]
• It is a Catalan number.^[3] 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.^[4] 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.^[5]
• 42 is a (2,6)-perfect number (super-multiperfect), as σ²(n) = σ(σ(n)) = 6n.^[6]
• 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(K_C) = k(2g − 2) (Hurwitz’s automorphisms theorem)
• 42 is the sum of the first 6 positive even numbers.
• 42 is the last natural number below 100 that can be expressed as a sum of three cubes without a representation being found: {\displaystyle (-80538738812075974)^{3}+80435758145817515^{3}+12602123297335631^{3}}.^[7]
• 42 is a Harshad number in base 10, because the sum of the digits 4 and 2 is 6 (4 + 2 = 6), and 42 is divisible by 6.
• 42 is the number of ways to arrange the numbers 1 to 9 in a 3×3 matrix such that the numbers in each row and column are in ascending order.
• 42 is also ten factorial divided by the number of seconds in a day (i.e. 86400).

Yeah, what about it!

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