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Magic Constant


The number

M_2(n)=1/nsum_(k=1)^(n^2)k
(1)
=1/2n(n^2+1)
(2)

to which the n numbers in any horizontal, vertical, or main diagonal line must sum in a magic square. The first few values are 1, 5, 15, 34, 65, 111, 175, 260, ... (OEIS A006003). The magic constant for an nth order magic square starting with an integer A and with entries in an increasing arithmetic series with difference D between terms is

 M_2(n;A,D)=1/2n[2A+D(n^2-1)]
(3)

(Hunter and Madachy 1975, Madachy 1979). In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to M_2(n).

For a magic cube, magic tesseract, etc., the magic d-D constant is

M_d(n)=1/(n^(d-1))sum_(k=1)^(n^d)k
(4)
=1/2n(n^d+1).
(5)

The first few magic constants are summarized in the following table.

nM_2(n)M_3(n)M_4(n)
SloaneA006003A027441A021003
1111
25917
31542123
434130514
5653151565

There is a corresponding multiplicative magic constant for multiplication magic squares.

A similar magic constant M_n^((j)) of degree k is defined for magic series and multimagic series as 1/n times the sum of the first n^2 kth powers,

M_n^((k))=1/nsum_(i=1)^(n^2)i^k
(6)
=(H_(n^2)^((-p)))/n,
(7)

where H_n^((k)) is a harmonic number of order k. The following table gives the first few values.

nk=1k=2k=3k=4
SloaneA006003A052459A052460A052461
11111
251550177
315956755111
434374462460962
565110521125430729

See also

Magic Cube, Magic Geometric Constants, Magic Hexagon, Magic Series, Magic Square, Multimagic Series, Multiplication Magic Square, Panmagic Square

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References

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 23-34, 1975.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 86, 1979.Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, 2002.Sloane, N. J. A. Sequences A006003/M3849, A021003, A027441, A052459, A052460, and A052461 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Magic Constant

Cite this as:

Weisstein, Eric W. "Magic Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicConstant.html

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