Area Magic Squares

Area magic squares are an idea of William Walkington.

Geometrical Interpretation of matrices

Area Matrix Definition (PDF)
An Area Matrix (A,M) is a combination of a positive matrix A and a set M of continuous graphs.
An Area Magic Square (AMS) is an area matrix (A,M) where the matrix A is a magic square.

Less mathematical definitions (PDF)

Linear Area Magic Squares of order 3

Third-Order Linear Area Magic Squares - Algorithm (PDF)

The following AMS were calculated with this algorithm. The graphics were made in high resolution by William Walkington.
Shown is each of the two solutions for magic 3x3 squares with consecutive numbers.
Click on a small picture to see it in a new frame, then click again to enlarge it.

	1 to 9	2 to 10	3 to 11	4 to 12	5 to 13	6 to 14	7 to 15	8 to 16
Solution 1
Solution 2

2017-01-06: First Third-Order Linear Area Magic Square (PDF)

Theory of area magic squares (PDF)

Third-Order Integer Area Square (PDF)

Curved Area Magic Square (PDF)

Linear Area Magic Squares of order 4

2017-01-14: First Order-4 Linear Area Magic Square

Article: Semi-orthogonal linear area magic squares of order 4 - part 1 (PDF)

1 What is a linear area magic square?
2 When were linear area magic squares first found?
3 How to present the number matrix?
4 What are the properties of the number matrix?
5 How many numbers or parameters are sufficient?
6 How can we describe all numbers by parameters?
7 How to avoid duplicates?
8 How to construct all L-AMS for a certain magic sum S?
9 What can be calculated with a computer?

Article: Semi-orthogonal linear area magic squares of order 4 - part 2 (PDF)

How to draw a semi-orthogonal linear area magic square of order 4?
How can we determine the distances?
Several examples.

Data: All 720 order-4 linear area magic squares up to S=200 (TXT)

Examples (animated GIF)

equivalent type 1

equivalent type 2

magic sum 144

Other Area Magic Squares of order 4

Classical Magic Square
2017-02-26 (PDF)

Consecutive Numbers
2017-02-02 (PDF)

Area Magic Squares of order 5

Semi-orthogonal linear area magic squares of order 5 do not exist. (Equal areas would occur)
If linear area magic squares of order 5 exist, they would consist of large entries.

Semi-orthogonal convex AMS
2017-02-09

Rounded linear AMS
2017-02-15 (PDF)

Many other magic area squares (also orders > 4) were found by William Walkington and Hans-Bernhard Meyer.

Links
Hans-Bernhard Meyer:	Observations on 4x4 Area Magic Squares with vertical Lines
William Walkington:	Area Magic Squares and Tori of order 3 Area Magic Squares of order 6 Area Magic Squares of order 4
Carlos Rivera:	Area Magic Squares with prime numbers (solutions by Jan van Delden)

Home: Notes on Magic Squares