# The Anti-Magic Square Project: Background

## Magic Squares

## Problems on Anti-Magic Squares

The AMS problem is one of many variations on Magic squares. People like to
change the problem to make things interesting. For example, one can choose to
use special entries to build one's square such as only primes. Or, you can
require additional properties such as taking the square of each entry is also a
magic square. The Anti-Magic problem is just one of these variations. Here are
some references to it in the literature:

An *anti-magic square* {a_{i,j}} of order *n*
is a square array of order *n* which possesses the property that the
set of all the sums of *n* numbers in every row, in every column and in
each diagonal consists of consecutive numbers.
**Problem:** Find a method of constructing an anti-magic square of every
order.

Abe, G. ``Unsolved Problems on Magic Squares.''

Disc. Math.
127, (1994) 3-13.

Many areas of research are in need of study, including:
- What are the general properties of antimagic squares?
- Are there any systematic methods by which antimagic
square may be constructed? ...
- Can small antimagic square be converted into larger ones?
- Can magic square be converted into antimagic squares by some
routine manipulation of the integers? ...
- Can antimagic squares be used in some practical manner? Magic
squares have found application in the fields of marketing, agriculture
statistical and atomic research.

Madachy, J. S. ``Magic and Antimagic Squares.''

Ch. 4 in
Madachy's Mathematical Recreations. New York: Dover, pp. 103-113, 1979.

Composed by John Cormie Updated: July 21 1999

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