magic square 327

There are no 2 ×2 magic squares of standard type,

just one standard 3 ×3 magic square, 880 standard

4×4 magic squares, and 275,305,224 standard 5 ×5

magic squares. To this day, no one knows the count of

6 ×6 standard magic squares.

General methods are known for constructing stan-

dard n×nmagic squares of any size nlarger than 2.

For example, for any odd value of n, begin by placing

the number 1 at any location inside the array and

incrementally placing subsequent numbers in the

square diagonally above and to the right. Follow a

“wrap-around effect” so that paths leading off the top

of the array return to the bottom, and those leading

off to the right return to the left. When one encounters

a square that is already filled, place the subsequent

integer in the cell directly below the current cell to

then continue on moving diagonally upward to the

right. For example, starting with 1 in the center, this

method produces the following 5 ×5 magic square.

(Notice, as one reads through the sequence of entries,

the numbers 6, 11, 16, and 21 were each “bumped”

down to a lower diagonal.)

This method of construction is known as the Siamese

method and is attributed to the French ambassador

to Siam (now Thailand) Simon de la Loubere (ca.

1670). Methods for constructing standard magic

squares of even order do exist, but are considerably

more complicated.

There are a plethora of alternative requirements

one could place on a square arrangement of numbers

to produce magic squares with alternative remarkable

properties. We list here just a few examples:

1. Semi-Magic Squares: A square array of numbers

that fails to be a magic square only because its

main diagonals do not add to the magic constant

is called a

SEMI

-

MAGIC SQUARE

. These squares

have the remarkable property that, when regarded

as matrices, the

MATRIX

sum, product, and inverse

of any collection of semi-magic squares is again

semi-magic.

2. Magic Multiplication Squares: A square array of

numbers in which every row, column, and diagonal

has the same product is said to be a magic multipli-

cation square. The following array, for instance, is

such a magic square.

3. Magic Division Squares: A square array of numbers

for which for each triple of numbers a, b, and cin a

row, column, or diagonal, the quotient a÷(b÷c) is

the same as a magic division square. For example,

the following array is such a square:

4. Addition-Multiplication Magic Squares: A square

that is simultaneously a magic square under addi-

tion and under multiplication is called an addition-

multiplication magic square. The following 8 ×8

array is an example of such a square: