## magic squareAn n × n square of the distinct whole numbers 1, 2, ..., n^{2}, such that the sum of the numbers along any row, column,
or main diagonal is the same. This sum is known as the magic constant and is equal to ½ n(n^{2} + 1). There is only one
3 × 3 magic square (not counting reflections and rotations), which
was known to the Chinese as long ago as 650 BC as lo-shu and is bound up with a variety of myths. Associations between
magic squares and the supernatural are also evident in early Indian and
Arabian mathematics. The 3 × 3 square can be written as:
Each row, column, and main diagonal sums to 15. If the rows are read as three-digit numbers, forwards and backwards, and then squared, we find the interesting relation 8162 + 3572 + 4922 = 6182 + 7532 + 2942. The reader may wish to see if the same rule holds for the columns and main diagonals.
n = 3, 4, 5, 6, 7, 8, and 9, which he associated with the seven "planets"
then known (including the Sun and the Moon). Albrecht Dürer's
famous engraving of Melancholia (1514) includes a picture of an order-4
magic square. There are 880 distinct squares of order-4 and 275,305,224
squares of order-5, but the number of larger squares is unknown. A square
that fails to be magic only because one or both of the main diagonal sums
don't equal the magic constant is called a semi-magic
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 pandiagonal square (also known as a panmagic or diabolical square). Pandiagonal squares exist for all
orders except 6, 10, 14, ..., 2(2i + 1). There are 48 pandiagonal
4 × 4 squares. If replacing each number n by its
square _{i}n_{i}^{2} produces another magic square, the
square is said to be a bimagic or doubly magic square. If a square is magic for n, _{i}n_{i}^{2},
and n_{i}^{3}, it is known as a trebly magic square.
n^{2} of the n × n magic square in the diagonals. (3) Relocate
any number not in the n × n square to the opposite hole
inside the square.An antimagic square is an n × n array
of integers from 1 to n^{2} in which each row, column, and
main diagonal produces a different sum such that these sums form a sequence
of consecutive integers. There are no antimagic squares of size 2 × 2 and 3 × 3 but plenty
of them for larger sizes. A 4 × 4 antimagic square is a square arrangement
of the numbers 1 to 16 so that the totals of the four rows, four columns,
and two main diagonals form a sequence of ten consecutive integers, for
example:
The principle of magic squares can be extrapolated from two dimensions to any number of higher dimensions, including magic cubes and magic tesseracts, whose cross-sections consist of magic cubes, and so forth. ## References- Andrews, William S.
*Magic Squares and Cubes*. Mineola, NY: Dover Publications, 1960. Second Edition. - Hendricks, John R. "Magic Tesseracts and N-Dimensional Magic Hypercubes."
*Journal of Recreational Mathematics*, 6 (3) (Summer 1973). - Hendricks, John R. "Black and White Vertices of a Hypercube."
*Journal of Recreational Mathematics*, 11(4), 1978-79. - Hendricks, John R. "Magic Hypercubes." Winnipeg, Canada: self published pamphlet, 1988.
## Related entries• Latin square• magic tour • magic cube ## Related category• GAMES AND PUZZLES | ||||||||||||||||||||||||||||||||||

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