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The 4x4 Pan-Magic Squares

0 0 1 1
1 1 0 0
0 0 1 1
1 1 0 0
0 1 1 0
1 0 0 1
0 1 1 0
1 0 0 1
0 1 0 1
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1

Discovery: Only Three Squares, Only One Pattern.

Early work by others was devoted to enumerating the number of possible magic squares. By contrast, the discovery here is that all the different order-4 pan-magic squares are variations on just three possible squares and these three are in turn based on one single underlying pattern. These conclusions were reached in 1998. If anyone knows of earlier work reaching the same conclusions, I would appreciate notification.

Summary

All of the Order 4 Pan-Magic Squares are based on the same underlying "Magic Carpet", a simple pattern consisting of alternating pairs of ones with zeros. Four samples of this pattern are multiplied by 8, 4, 2, and 1, to make the Magic Carpets which are added together to make the final square. The 8, 4, 2, and 1, can be used in any order to make different squares.

  1. All order 4 pan-magic squares can be decomposed into four pan-magic carpets.
  2. Each carpet is obtainable from the same larger pan-magic carpet
  3. Substituting 1, 2, 4, and 8, into the carpets offers 24 potential permutations.
  4. In practice this produces only three truly different pan magic squares of order four.

Example of Magic Carpet Construction.

This sequence of squares show the production of one square by this technique:

0 0 8 8
8 8 0 0
0 0 8 8
8 8 0 0
+
0 4 4 0
4 0 0 4
0 4 4 0
4 0 0 4
+
0 2 0 2
0 2 0 2
2 0 2 0
2 0 2 0
+
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1
=
0 7 12 11
13 10 1 6
3 4 15 8
14 9 2 5

Latin Squares and Other Alphabetical Patterns.

The 4x4 Pan-Magic Squares are associated with interesting Alphabetical Squares and Patterns. This information is put on a separate page to save space here.

Horiz Bar

Only Three Possible Pan-Magic Squares

Only Three Squares.

Because Latin squares make only a limited contribution to understanding order 4 pan-magic squares, the magic squares are analyzed here in numerical form. Although the four examples of magic carpets above might yield 24 different Pan-Magic using all the possible substitutions of 8, 4, 2, and 1, in fact, translocation, rotation and reflection actually reduces the twenty-four potential squares to only three.

Standard Orientation.

To check this, every square was rotated, reflected and translocated using the same rules: put the zero at top left; put the next lowest possible number (always 7) is beside the zero; then reflect if necessary so that the square beneath the zero is the lowest of the two options. With the squares so arranged, inspection confirms that only three truly different squares are possible. The 24 are made up of the eight copies of the three basic squares produced by rotation and reflection.

The 3 Regular 4x4 Pan-Magic Squares

0 7 9 14
11 12 2 5
6 1 15 8
13 10 4 3
"S" = 8
"A" = 1
"N" = 4
"C" = 2
0 7 10 13
11 12 1 6
5 2 15 8
14 9 4 3
"S" = 8
"A" = 2
"N" = 4
"C" = 1
0 7 12 11
13 10 1 6
3 4 15 8
14 9 2 5
"S" = 8
"A" = 4
"N" = 2
"C" = 1

The Four Small Magic Carpets.

Note: each square has been assigned a name to
roughly correspond to the red squares - the "1"s.

0 0 1 1
1 1 0 0
0 0 1 1
1 1 0 0
S
0 1 1 0
1 0 0 1
0 1 1 0
1 0 0 1
A
0 1 0 1
0 1 0 1
1 0 1 0
1 0 1 0
N
0 1 0 1
1 0 1 0
1 0 1 0
0 1 0 1
C

Derivation of the Three Possible Squares.

The table above shows how the four small carpets make the three possible squares. The values 8, 4, 2, and 1, are substituted in the carpets according to the legend under each Magic square.

The 384 Pan-Magic Squares

From these three basic squares all of the other possible squares can be made. There are sixteen cells in each square so, just by translocation, there are sixteen variations of each square. Each of these translocations can be rotated into four different positions, and each rotation can be reflected. Therefore, 3 x 16 x 8 = 384 regular, order four, pan-magic squares can be derived from these three basic squares.

Historical Note

Attempts to provide complete listings of magic squares go back to at least 1693 when Frenicle in France concluded there were 880 squares of order 4. He was including all types of magic square - not just the pan-magic ones. This is reported by Henry Dudeney in his book which was first published in 1917; he claims that these results have been ". . . verified over and over again". On the next page Dudeney goes on to provide an analysis and classification which he attributes to "Mr. Frost" - one of the earlier students of magic squares. He describes "Nasik" squares - the only type which he describes as having properties we now call "Pan-Magic". ("Nasik" was the name of the town in India in which Mr. Frost lived.)

The example of the Nasik square shown by Dudeney corresponds to an example of Type 3 above. Dudeney states there are 48 Nasik Squares and adds that each square can produce seven more by "reversals and reflections". This number, 48, implies that he and other early workers might have failed to recognize that they were reproducing the 16 translocations of three basic squares - the ones described here.

The reduction to just three fundamental squares is, however, not mentioned by Dudeney, or by other readily accessible texts.


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Mar 6, 2010