An n × n square of the distinct whole numbers 1, 2, ..., n2, 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(n2 + 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
3572 + 4922 = 6182 + 7532 + 2942.
The reader may wish to see if the same rule holds for the columns and main
In the early 16th century Cornelius Agrippa constructed squares for 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 ni by its
square ni2 produces another magic square, the
square is said to be a bimagic or doubly magic square. If a square is magic for ni, ni2,
and ni3, it is known as a trebly magic square.
|An old Chinese magic square
A little trial and improvement is all it takes to construct the 3 ×
3 magic square, but for building 4 × 4 squares and larger, a systematic
method, or algorithm, is important. Interestingly,
different algorithms are needed depending in whether the square is of an
even order or an odd order. Odd order squares are the easier variety to
make and there are several standard techniques, including the Siamese (sometimes
called de la Loubere's or the Staircase), the Lozenge, and de Meziriac's
methods. Here is yet another approach, known as the Pyramid or extended
diagonals method: (1) Draw a pyramid of same size squares as the magic square's
squares, on each side of the magic square; the pyramid should be two less,
in number of squares on its base, than the number of squares on the side
of the magic square. (2) Sequentially place the numbers 1 to n2 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.
|A magic square of order 5
An antimagic square is an n × n array
of integers from 1 to n2 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
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.
- 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