Magic Constant
The number
to which the
numbers in any horizontal, vertical, or main diagonal line must sum in a Magic Square. Thefirst few values are 1, 5 (no such magic square), 15, 34, 65, 111, 175, 260, ... (Sloane's A006003). The magic constant for anth order magic square starting with an Integer 
and with entries in an increasing Arithmetic Series withdifference 
between terms is
(Hunter and Madachy 1975, Madachy 1979). In a Panmagic Square, in addition to the main diagonals, the broken diagonalsalso sum to
.For a Magic Cube, the magic constant is
The first few values are 1, 9, 42, 130, 315, 651, 1204, ... (Sloane's A027441).
There is a corresponding multiplicative magic constant for Multiplication Magic Squares.
See also Magic Cube, Magic Geometric Constants, Magic Hexagon, Magic Square, MultiplicationMagic Square, Panmagic Square
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. Sloane, N. J. A. SequencesA027441 andA006003/M3849in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
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