328 magnitude

5. Border Squares: A magic square (under addition)

that remains magic when its border is removed is

called a border magic square. If one can continue to

remove borders from subsequent squares and still

preserve the magic property, then we say we have a

nested magic square. The following is an example of

a nested magic square:

6. Alphamagic Squares: If replacing each entry in an

(additive) magic square with the number of letters in

the (English) name of that number produces a new

magic square, then we say that we have an alpham-

agic square. For example, the square below, written

in words, is alphamagic (The number of letters in

each name is written in parentheses, forming its own

magic square.)

7. Magic Rectangles: A rectangular array of numbers

in which the sum of entries in any row or column is

the same is called a magic rectangle. If there are n

rows and mcolumns, and the magic constant is M,

then summing all the rows shows that nM equals

the sum of all entries in the array. Summing all

columns gives nM as the sum of all entries in the

array, and so we must have nM = mM. If nand m

are required to be different, this forces the magic

constant to be zero. Thus any magic rectangle, such

as the one below, must include negative entries.

One can also consider alternative arrangements of

numbers and look for magic pentagrams and hexa-

grams, or magic cubical arrays of number, for instance.

This is a popular topic of exploration for general enthu-

siasts of mathematics.

magnitude A positive measure of the size of a quan-

tity is called its magnitude. For example, the magnitude

of a real number is its

ABSOLUTE VALUE

. (Thus, for

instance, 27 and –27, although of opposite polarity,