The answer is 42.
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 cubes whose sum of values is 42.
Forty-two is a pronic number and an abundant number; its prime factorization 2 · 3 · 7 makes it the second sphenic number and also the second of the form { 2 · 3 · r }. As with all sphenic numbers of this form, the aliquot sum is abundant by 12. 42 is also the second sphenic number to be bracketed by twin primes; 30 is also a pronic number and also rests between two primes. 42 has a 14 member aliquot sequence 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 and is itself part of the aliquot sequence commencing with the first sphenic number 30. Further, 42 is the 10th member of the 3-aliquot tree.
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 conjectured to be the scaling factor in the leading order term of the "sixth moment of the Riemann zeta function". In particular, Conrey & Ghosh have conjectured
where the infinite product is over all prime numbers, p.[1][2]
It is the third pentadecagonal number. It is a meandric number and an open meandric number.
42 is a Størmer number.
42 is a perfect score on the USA Math Olympiad (USAMO)[3] and International Mathematical Olympiad (IMO).[4]
In base 10, this number is a Harshad number and a self number, while it is a repdigit in base 4 (as 222).
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 [6]
