Animated Magic Square

What Is A Magic Square?

The first question, for those who have not come across Magic Squares before, must be, "What is a Magic Square?"

There are two types of Magic Square:
  1. Alphabetic
  2. Numeric.

Alphabetic Magic Squares

There are two types of Alphabetic Magic Square:
  1. Word Squares
  2. Latin Squares.

Word Squares

Word Square The early ones included a series of letters, arranged in a square, to spell certain words. One of the most well-known of this variety, throughout the Western world at any rate, is the one shown to the right:

Note that the five Latin words appear in the same order both horizontally and vertically, and that the five rows may also be read palindromically. They also form a sentence - "Arepo, the sower, guides the wheels at work".

Many people also think that it has a Christian significance, because it contains, jumbled up, the words "pater noster", which, of course, means "our father", together with the letters "a" and "o", which are the Greek letters Alpha and Omega, used of God, for example, in Revelations.

These sorts of Word Square frequently occur in puzzle magazines or you can find them on new bingo sites today.

Latin Squares

Latin Square Latin Squares are usually formed with "n" sets of letters, where "n" is the number of cells on each side of the square. They should be arranged so that no horizontal row or vertical column contains the same two letters.

If the two corner diagonals also contain no repeated letters, then it is called a Diagonal Latin Square. The following is an example of such a square:

Latin Squares may also be composed using numbers or any other symbols (e.g. ESP symbols, road signs).

Numeric Magic Squares

Numeric Magic Squares are much more interesting because of their mathematical properties. This is because a Numeric Magic Square must consist of a series of numbers arranged in a square in such a manner that the sum of each row, each column and of both the corner diagonals adds up to the same amount, which is called the "Magic Total".

3x3 Magic Square The animated Magic Square at the top of this page shows three different Numeric Magic Squares, of "orders" three, four and five respectively.

If you are unable to view this animated version, because of the Browser you are using, here is the most well-known Magic Square, of "order 3":

Numeric Magic Squares may be divided into two categories:

  1. "Odd" Magic Squares, which means that there is an odd number of cells on each side of the Magic Square.
  2. "Even" Magic Squares, which means that there is an even number of cells on each side of the Magic Square. "Even" Magic Squares may be further divided into two sub-categories:
    • "singly even" Magic Squares, which means that the number of cells on each side of the Magic Square is evenly divisible by two, but not by four (e.g. 6 x 6 and 10 x 10 Magic Squares)
    • "doubly even" Magic Squares, which means that the number of cells on each side of the Magic Square is evenly divisible by both two and four (e.g. 4 x 4 and 8 x 8 Magic Squares).
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This page is Copyright © 1997, Mark S. Farrar.
Created: Monday 29th December, 1997

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