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The 4x4 Latin Squares and Alphabetical Patterns

Patterns and 4x4 Alphabetical Squares.

2x2 pattern

The fundamental 4x4 magic carpet can be represented by dots and spaces. Any line in any direction of length four contains two dots, i.e., it sums to 2; any selected 4x4 area is a pan-magic pattern. We can take four samples of this large carpet, rotate two of them, and make the only four possible order four magic carpets.

Pattern S
Pattern A
Pattern N
Pattern C

Alphabetical Subsitution.

Because they look somewhat like letters of the alphabet they are given a letter to identify them.

Alpha 4x4

The Composite Alphabetical Magic Carpet

The dot in each of the above squares is replaced with its own letter. The four squares are then combined to make the composite square on the left and then the larger carpet on the right.


Only one composite pattern exists. This one composite square underlies all order 4 pan-magic squares. Any 4x4 area contains each letter twice in every row, every line, and every diagonal. To make an actual 4x4 magic square the letters in this square would be replaced, respectively by 8, 4, 2, and 1 (see Main 4x4 Page).


Neat Pattern.

When the pattern is repeated, a large magic carpet emerges - pleasing and symmetrical - which makes the interesting colored pattern on the left.

Not a Latin Square

Strictly speaking this is not a "Latin Square". A Latin Square for order N uses N letters N times and each row and each column contains one of each letter. The above square can be converted into two Latin Squares.

Two 4x4 Latin Squares

The illustration below combines two 4x4 Latin Squares into a single, so-called Graeco-Latin square For convenience it employs Upper and Lower case Roman letters instead of using both Roman and Greek characters. The letters in the new square are derived from the square above:

A when neither S nor N are present
B when N is present
C when S is present
D when both S and N are present
  a when neither C nor A are present
b when C is present
c when A is present
d when both C and A are present

a d c b
d a b c
b c d a
c b a d
Aa Bd Cc Db
Cd Da Ab Bc
Bb Ac Dd Ca
Dc Cb Ba Ad

Not Pan-Magic.

Inspection of the resulting Graeco-Latin shows that that the rows and columns are inevitably "Magic" - they do contain one of each letter. This is, however, not true for any of the diagonals; they can only add up to the magic sum for appropriately selected numerical substitutions.

Limited Value for 4x4 Latin Squares.

Because of this limitation, Latin Squares are of only limited use in constructing 4x4 pan-magic squares. There are two other possible 4x4 alphabetical squares and all three are shown below. The third is not even Latin in that, now, only the diagonals contain one of each letter.

Aa Bd Cc Db
Cd Da Ab Bc
Bb Ac Dd Ca
Dc Cb Ba Ad
Aa Bd Cb Dc
Cd Da Ac Bb
Bc Ab Dd Ca
Db Cc Ba Ad
Aa Bd Da Cd
Db Cc Ab Bc
Ad Ba Dd Ca
Dc Cb Ac Bb

Magic Carpets vs. Latin Squares.

Regular prime number pan magic squares (larger then order 3), e.g., order 5, are composed of two Magic Carpets, or Latin Squares. For such squares the Latin Square and the Magic Carpet are one and the same. Neither of the two Latin Squares can be decomposed further into more detailed Magic Carpets, and all rows, columns, and diagonals contain one of each letter.

However, for many other magic squares, the Latin Squares are readily, and helpfully, decomposed into the component magic carpets, e.g., order 4 is best understood as four binary magic carpets; order 8 as 6 binary magic carpets; and order 9 as 4 magic carpets in base 3. These component magic carpets use each letter more than once in each line - hence they are not Latin Squares. The term "Magic Carpets" embraces both these valuable patterns as well as the "Latin Squares", i.e., all Latin Squares are Magic Carpets, but not vice-versa.

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Copyright © Mar 2010 Magic Squares
Mar 6, 2010