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The 3x3 (and Associated) Magic Squares

A Method for Creating 4p+2 Magic Squares

Introduction. The 3x3 Magic Square is included because it is simple enough to understand readily and because it is used for making Magic Squares of Order 6. 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.

The 3x3.

There is just one 3x3 magic square although rotations and reflections produce eight variations. 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 is used to construct 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. Because the underlying carpet is not Pan-Magic it is inevitable that the resulting Magic square is not going to be pan-magic either. This is becasue one of the carpet diagonals has to consist of the numeral one repeated three times. As a result one of the broken diagonals has to be three zeros, and the other has to be three twos.

3 x
 2 0 1 0 1 2 1 2 0
+
 1 2 0 0 1 2 2 0 1
=
 7 2 3 0 4 8 5 6 1

Making the 6x6, the 10x10, and larger.

The associated pages show how such a simple square can be combined with a magic grid to make the members of the order 4p+2 magic squares.

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