The 6x6 Magic Squares

Around 1789, Euler formulated his "conjecture" that there were no Graeco-Latin squares of orders 2, 6, 10, 14, etc. Gaston Tarry proved in 1901 that there were no Graeco-Latin square of order 6, which supported Euler's conjecture. Then, in 1959, Parker, Bose and Shrikhande constructed an order 10 Graeco-Latin square, and showed a construction technique for orders 10, 14, 18, etc. They therefore disproved Euler's conjecture, leaving the 6x6 magic square as something of an oddity - it cannot be based on a pair of identical latin squares.

A Method for Creating 4p+2 Magic Squares

What is presented here is a simple, logical method of constructing a 6x6 Magic square. The structure is based on the 3x3 magic square combined with small 2x2 cells. A similar method was first described by Edward Falkener in 1892 on page 294 of his book Games, Ancient and Oriental

The The 3x3 page showed how two component carpets could be used to make a 3x3 magic square:

Making the 6x6.

Two different types of component are used. The first is the pattern in the two 3x3 carpets above. Each one is doubled in size by duplicating each number in both axes. The two resulting squares are summed to produce one of the required components.

Magic Carpet.

The second component is a magic carpet composed of a simple pattern. Two versions are required, the second merely being a rotation of the first. Combined they make a magic carpet which distribute the numbers 0, 1, 2, and 3 so that one of each appears in every one of the nine blocks of similar numbers in the "First Component".

When the two Component squares are added together they make the 6x6 magic square above with a Magic Constant of 105.

Twenty Four Possible Combinations

The above four Magic Carpets (A, B, C & D) were multiplied by 12, 4, 2, and 1 to arrive at the final 6x6 magic square . This sequence of multipliers is just one of the 24 which are capable of producing a consecutive series commencing at zero and finishing at 35. The list on the right shows all the possible combinations:

Each different row of multipliers produces a different square - twenty four in all. These are shown below. The number above each square corresponds to the row in this list. Without knowing the construction, it would not be immediately obvious that these squares share a common underlying structure.

24 Examples from 1 Source

1                   2                   3                   4
22 21 24 25  6  7   21 22 24 26  5  7   17 22 24 30  3  9   22 17 24 25  8  9
20 23 27 26  5  4   20 23 27 25  6  4   16 23 31 25  8  2   16 23 31 30  3  2
3  0 17 16 35 34    3  0 18 16 35 33    7  0 20 14 35 29    7  0 15 14 35 34
1  2 19 18 33 32    2  1 19 17 34 32    6  1 21 15 34 28    1  6 21 20 29 28
31 30  8  9 12 15   31 29  8 10 12 15   33 27  4 10 12 19   33 32  4  5 12 19
28 29 10 11 14 13   28 30  9 11 13 14   26 32  5 11 13 18   26 27 10 11 18 13

5                   6                   7                   8
20 17 24 27  7 10   17 20 24 30  4 10   30 29  8  9 14 15   29 30  8 10 13 15
14 23 33 30  4  1   14 23 33 27  7  1   28 31 11 10 13 12   28 31 11  9 14 12
9  0 16 13 35 32    9  0 19 13 35 29    3  0 17 16 35 34    3  0 18 16 35 33
3  6 22 19 29 26    6  3 22 16 32 26    1  2 19 18 33 32    2  1 19 17 34 32
34 31  2  5 12 21   34 28  2  8 12 21   23 22 24 25  4  7   23 21 24 26  4  7
25 28  8 11 18 15   25 31  5 11 15 18   20 21 26 27  6  5   20 22 25 27  5  6

9                  10                  11                  12
32 27  4  5 18 19   27 32  4 10 13 19   31 28  2  5 18 21   28 31  2  8 15 21
26 33 11 10 13 12   26 33 11  5 18 12   25 34 11  8 15 12   25 34 11  5 18 12
7  0 15 14 35 34    7  0 20 14 35 29    9  0 16 13 35 32    9  0 19 13 35 29
1  6 21 20 29 28    6  1 21 15 34 28    3  6 22 19 29 26    6  3 22 16 32 26
23 22 24 25  2  9   23 17 24 30  2  9   23 20 24 27  1 10   23 17 24 30  1 10
16 17 30 31  8  3   16 22 25 31  3  8   14 17 30 33  7  4   14 20 27 33  4  7

13                  14                  15                  16
23 14  6 15 19 28   25 16  2 11 21 30   26 11 12 15 19 22   31 16  2  5 24 27
5 32 33 24 10  1    7 34 29 20 12  3    8 29 33 30  4  1   13 34 23 20  9 6
27  0 13  4 35 26   27  0 13  4 35 26   21  0 10  7 35 32   21  0 10  7 35 32
9 18 31 22 17  8    9 18 31 22 17  8    3 18 28 25 17 14    3 18 28 25 17 14
34 25  2 11  3 30   32 23  6 15  1 28   34 31  2  5  6 27   29 26 12 15  1 22
7 16 20 29 21 12    5 14 24 33 19 10   13 16 20 23 24  9    8 11 30 33 19  4

17                  18                  19                  20
28 11 12 13 20 21   32 15  4  5 24 25   14 23  6 24 10 28   16 25  2 20 12 30
10 29 31 30  3  2   14 33 23 22  7  6    5 32 33 15 19  1    7 34 29 11 21  3
19  0  9  8 35 34   19  0  9  8 35 34   27  0 22  4 35 17   27  0 22  4 35 17
1 18 27 26 17 16    1 18 27 26 17 16   18  9 31 13 26  8   18  9 31 13 26  8
33 32  4  5  6 25   29 28 12 13  2 21   34 16  2 20  3 30   32 14  6 24  1 28
14 15 22 23 24  7   10 11 30 31 20  3    7 25 11 29 12 21    5 23 15 33 10 19

21                  22                  23                  24
11 26 12 30  4 22   16 31  2 20  9 27   11 28 12 30  3 21   15 32  4 22  7 25
8 29 33 15 19  1   13 34 23  5 24  6   10 29 31 13 20  2   14 33 23  5 24  6
21  0 25  7 35 17   21  0 25  7 35 17   19  0 26  8 35 17   19  0 26  8 35 17
18  3 28 10 32 14   18  3 28 10 32 14   18  1 27  9 34 16   18  1 27  9 34 16
34 16  2 20  6 27   29 11 12 30  1 22   33 15  4 22  6 25   29 11 12 30  2 21
13 31  5 23  9 24    8 26 15 33  4 19   14 32  5 23  7 24   10 28 13 31  3 20

Twenty Four Magic Squares from a single "Root" Pattern

Each of these 24 basic squares can be rotated and reflected to produce eight derived magic squares, i.e., the one original set of root patterns yields 24 x 8 = 192 magic squares. This far from the complete number of order six magic squares. There are other techniques for constructing them, but this technique is easy to visualize, easy to understand, and is applicable to other sizes.

Making the 10x10, and larger.

The 10x10 page show how this simple technique can be applied to larger members of the order 4p+2 magic squares.