The 3x3 (and associated) Magic Squares

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Magic Squares
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Index: 		Magic Carpets || Make Your Own || How Many? || Spreadsheets
The Formula || Techniques || History || References || Downloads
Individual Magic Squares by Order: 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13
Regular Prime Pan-Magic Squares: 5 || 7 || 11 || 13
Multiple Carpet Pan-Magic Squares: 4 || 8 || 9 || 12
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4P+2 Magic Squares (6, 10 etc.): 3 || 6 || 10

A Method for Creating 4p+2 Magic Squares

Introduction. These pages describe an elegant technique for making Magic Squares of Order 6, 10, 14, etc. The method combines two techniques:
  1. doubling the size of an odd order squares, and using this in conjunction with
  2. using a magic grid once in one axis and, again, rotated in the other axis.
Early researchers regarded the 6x6 magic square as difficult or impossible to construct. It may have been because these squares did not yield to the logical construction which applies to the pan-magic squares of orders 4 and 5. Or, yet again, it may have been bacause, although there are magic squares, there are no pan-magic squares of order 6, 10, 14, (4p+2) etc. The smallest of these squares is based on the 3x3 magic square.

The 3x3. There is just one 3x3 magic square although, with rotations and reflections, there are eight variations of what is essentially the same squares. The 3x3 square cannot be pan-magic. It is included here so that visitors can find an example and because it is used to create the 6x6 magic square. A similar procedure will be used to costruct other squares of the order N = 4p +2, i.e., 6, 10, 14, etc.

The Two Carpets. The squares below indicate how two identical carpets (one rotated) produce the only possible 3x3. Inevitably, the 3x3 is not going to be pan-magic. One of the main diagonals has to consist of the numeral one repeated three times. As a result one of the two parallel, broken, diagonals has to be three zeros, and the other has to be three twos. Therefore, these patterns cannot make "pan-magic carpets" and the resulting square cannot be pan-magic.

3 x

1 2 0
0 1 2
2 0 1

+
2 0 1
0 1 2
1 2 0

=
5 6 1
0 4 8
7 2 3


Making the 6x6, the 10x10, and larger. The associated pages show how such a simple square can be combines with a magic grid to make the members of the order 4p+2 magic squares.

Magic Squares Home
Website Home Page		 Topic  
Index: 		Magic Carpets || Make Your Own || How Many? || Spreadsheets
The Formula || Techniques || History || References || Downloads
Individual Magic Squares by Order: 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13
Regular Prime Pan-Magic Squares: 5 || 7 || 11 || 13
Multiple Carpet Pan-Magic Squares: 4 || 8 || 9 || 12
4P+2 Magic Squares (6, 10 etc.): 3 || 6 || 10
Select Version: Frames || NO Frames (faster)

Magic Squares
Home Page
Topic  
Index: 		Magic Carpets || Make Your Own || How Many? || Spreadsheets
The Formula || Techniques || History || References || Downloads
Individual Magic Squares by Order: 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13
Regular Prime Pan-Magic Squares: 5 || 7 || 11 || 13
Multiple Carpet Pan-Magic Squares: 4 || 8 || 9 || 12
Website
Home Page
4P+2 Magic Squares (6, 10 etc.): 3 || 6 || 10