Magic constant

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For unnamed numerical constants, see Magic number (programming)#Unnamed numerical constants.

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general M = n(n^2 +1)/2.

MagicSquare-ExplicitSums.png


Harvey Heinz has a reference to a 8 x 8 magic square that Woodruff made in 1916.[1] Each 8 cell segment of the Peano space filling curve sums to the magic constant. The second square is one of Dame Kathleen Ollerenshaw's 368,640 8 x 8 most perfect magic squares that also meet the above criteria. [2]

1 32 34 63 37 60 6 27
48 49 15 18 12 21 43 54
19 14 52 45 55 42 24 9
62 35 29 4 26 7 57 40
25 8 58 39 61 36 30 3
56 41 23 10 20 13 51 46
11 22 44 53 47 50 16 17
38 59 5 28 2 31 33 64
36 21 4 53 10 63 42 31
25 48 57 16 51 6 19 38
52 5 20 37 26 47 58 15
9 64 41 32 35 22 3 54
55 2 23 34 29 44 61 12
14 59 46 27 40 17 8 49
39 18 7 50 13 60 45 28
30 43 62 11 56 1 24 33
Hilbert curve.svg


The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes.

Magic stars[edit]

The magic constant of an n-pointed normal magic star is M = 4n + 2.

Magic series[edit]

In 2013 Dirk Kinnaes found the magic series polytope. The number of unique sequences that form the magic constant is now known up to n=1000.[3]

Physics[edit]

In the mass model the value in each cell specifies the mass for that cell.[4] This model has two notable properties. First it demonstrates the balanced nature of all magic squares. If such a model is suspended from the central cell the structure balances. ( consider the magic sums of the rows/columns .. equal mass at an equal distance balance). The second property that can be calculated is the moment of inertia. Summing the individual moments of inertia ( distance squared from the center x the cell value) gives the moment of inertia for the magic square. "This is the only property of magic squares, aside from the line sums, which is solely dependent on the order of the square" [5]

See also[edit]

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