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Magic Square |  |
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A (normal) magic square consists of the distinct positive
integers 1, 2, ...,  The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order 4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown above.
The magic
constant for an  (Hunter and Madachy 1975). If every number in a magic square is subtracted from  ,
another magic square is obtained called the complementary magic
square. Squares which are magic under multiplication instead of
addition can be constructed and are known as multiplication
magic squares. In addition, squares which are magic under both
addition and multiplication can be constructed and are known
as addition-multiplication
magic squares (Hunter and Madachy 1975).
A square that fails to be magic only because one or both of the
main diagonal sums do not equal the magic
constant is called a semimagic
square. If all diagonals (including those obtained by
wrapping around) of a magic square sum to the magic
constant, the square is said to be a panmagic
square (also called a diabolic
square or pandiagonal
square). If replacing each number
 Kraitchik (1942) gives general techniques of constructing even and odd squares
of order
A generalization of this method uses an "ordinary vector"
 A second method for generating magic squares of odd order has been discussed by J. H. Conway under the name of the "lozenge" method. As illustrated above, in this method, the odd numbers are built up along diagonal lines in the shape of a diamond in the central part of the square. The even numbers which were missed are then added sequentially along the continuation of the diagonal obtained by wrapping around the square until the wrapped diagonal reaches its initial point. In the above square, the first diagonal therefore fills in 1, 3, 5, 2, 4, the second diagonal fills in 7, 9, 6, 8, 10, and so on.
 An elegant method for constructing magic squares of doubly
even order
 A very elegant method for constructing magic squares of singly
even order
It is an unsolved problem to determine the number of magic
squares of an arbitrary order, but the number of distinct magic
squares (excluding those obtained by rotation and reflection) of
order
 The above magic squares consist only of primes and were discovered by E. Dudeney (1970) and A. W. Johnson, Jr. (Gardner 1984, p. 86; Dewdney 1988). Madachy (1979, pp. 93-96) and Rivera discuss other magic squares composed of primes.
 Benjamin Franklin constructed the above
In addition to other special types of magic squares, a
According to a 1913 proof of J. N. Murray (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive primes starting with 3 and including the number 1 is of order 12.
Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the alphamagic square and templar magic square.
Various numerological properties have also been associated with magic squares. Pivari associates the squares illustrated above with Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, respectively. Attractive patterns are obtained by connecting consecutive numbers in each of the squares (with the exception of the Sun magic square).
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such that the sum of the 
numbers in any horizontal, vertical, or main diagonal line is
always the same 
th order magic square starting with an 
and with entries in an increasing 
between terms is 
by its square 
produces another magic square, the
square is said to be a 
,
, and 
,
it is called a 
, the square is said to be an 
. For 
which gives the offset for each noncolliding move and a
"break vector" 
which gives the offset to introduce
upon a collision. The standard Siamese method therefore has ordinary
vector (1, 
and break vector (0, 1). In order
for this to produce a magic square, each break move must end up on
an unfilled cell. Special classes of magic squares can be
constructed by considering the absolute sums 
,
, 
,
and 
. Call the set of these numbers the sumdiffs
(sums and differences). If all sumdiffs are 
and the square is a magic square,
then the square is also a 
)
)
)
)
is to draw 
s
through each 
subsquare and fill all squares in
sequence. Then replace each entry 
on a crossed-off diagonal by 
or, equivalently, reverse the order of the crossed-out entries. Thus
in the above example for 
,
the crossed-out numbers are originally 1, 4, ..., 61, 64, so entry 1
is replaced with 64, 4 with 61, etc.
with 
(there is no magic square of order 2) is due to
J. H. Conway, who calls it the "LUX" method. Create an
array consisting of 
rows of 
s, 1 row of Us, and 
rows of 
s, all of length 
. Interchange the middle U with the L above it. Now
generate the magic square of order 
using the Siamese method centered on the array of letters (starting
in the center square of the top row), but fill each set of four
squares surrounding a letter sequentially according to the order
prescribed by the letter. That order is illustrated on the left side
of the above figure, and the completed square is illustrated to the
right. The "shapes" of the letters L, U, and X naturally suggest the
filling order, hence the name of the algorithm.
, 2, ... are 1, 0, 1, 880, 275305224, ...
(Sloane's 
squares is not known, but Pinn and Wieczerkowski (1998)
estimated it to be 
using Monte Carlo simulation and methods from statistical mechanics.
square whose entries are consecutive 
