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Magic Constant






The term magic constant or '''magic sum''' is similarly applied to other "magic" figures such as Magic Star s and Magic Cube s.


NORMAL MAGIC SQUARES


If a magic square of order ''n'' is Normal (i.e., it contains the numbers 1 to ''n''2), then the magic constant depends only on ''n''; its value is
:M_2(n) = rac{n(n^2+1)}{2}.

This formula is a consequence of the formula for the sum of the first n integers
:1 + 2 + ... + k = rac{k(k+1)}{2}
applied to the case ''k'' = ''n''2, yielding ''n''2(''n''2+1)/2, which is then divided by ''n'' because there are ''n'' rows, all of which sum to the same value.

The magic constants of normal magic squares of order ''n'' = 3, 4, 5, … are (sequence in OEIS ):

: 15 , 34 , 65 , 111 , 175 , 260 , 369, 505, 671, 870

The numbers in any row, column, or diagonal of a normal magic square form a Magic Series .


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