# Frierson

Like de la Hire, L.S. Frierson works with two auxiliary squares, which will be superimposed. Using the method de la Hire, the middle numbers are only arranged on the diagonals, while Frierson works with a geometric arrangement of these numbers that creates symmetrical magic squares.

• Place the median of the numbers 1, 2, 3, … , n in the center cell of the square. In the example of order n=5, this would be the number 3.
• The n − 1 remaining mean numbers are now arranged in pairs symmetrically to this number. However, it must be ensured that this number appears exactly once in every row and every column.
• The second auxiliary square is filled similar, where that the median of the first auxiliary square is replaced by the median of the sequence 0, n, 2n, 3n, …. In an example of order n=5 this would be the number 10.
• This auxiliary square is also reflected at the central vertical axis.
•  3 3 3 3 3
•  10 10 10 10 10
•  10 10 10 10 10

Now the center rows of the two auxiliary squares have to be filled.

• For the first auxiliary square, take the remaining numbers of the sequence 1, 2, 3, … , n and place them symmetrically to the center cell. So the sum of horizontally symmetrical pairs of numbers must always be n+1.
• Do the same with the remaining numbers in the sequence 0, n, 2n, 3n, … for the second auxiliary square. Again, the sum of the pairs of numbers must always be the same. Here the sum of each pair (n-1) · n.
•  3 3 4 1 3 5 2 3 3
•  10 10 0 15 10 5 20 10 10

Now you only have to fill in the remaining rows of the two auxiliary squares so that the numbers appear in the same relative order as in the center rows. A symmetrical magic square is then created by superimposing the two auxiliary squares, which simply means that corresponding cells are added.

•  1 3 5 2 4 3 5 2 4 1 4 1 3 5 2 5 2 4 1 3 2 4 1 3 5
•  20 0 15 10 5 5 20 0 15 10 0 15 10 5 20 10 5 20 0 15 15 10 5 20 0
•  21 3 20 12 9 8 25 2 19 11 4 16 13 10 22 15 7 24 1 18 17 14 6 23 5

This procedure can be applied to all higher orders. The following figure shows a magic square of order 7 constructed using this method.

•  3 6 4 2 5 1 7 2 5 1 7 3 6 4 6 4 2 5 1 7 3 7 3 6 4 2 5 1 5 1 7 3 6 4 2 4 2 5 1 7 3 6 1 7 3 6 4 2 5
•  7 35 14 42 21 0 28 21 0 28 7 35 14 42 28 7 35 14 42 21 0 35 14 42 21 0 28 7 42 21 0 28 7 35 14 0 28 7 35 14 42 21 14 42 21 0 28 7 35
•  10 41 18 44 26 1 35 23 5 29 14 38 20 46 34 11 37 19 43 28 3 42 17 48 25 2 33 8 47 22 7 31 13 39 16 4 30 12 36 21 45 27 15 49 24 6 32 9 40

This geometric method of Frierson is very universal and creates a variety of symmetrical magic squares. One of the reasons for this is that the numbers in the center row can be arranged in very different ways. The various geometrical arrangements from which one can start are of even greater importance. To give an idea of ​​the variety, the next figure shows some of the geometric arrangements for order n=7.

•  4 4 4 4 4 4 4
•  4 4 4 4 4 4 4
•  4 4 4 4 4 4 4

In the previous examples, the median were only placed in the upper left and lower right areas. Of course, the other two areas can also be used. This opens up a large variety of additional possibilities, of which a small selection of different arrangements is shown here.

•  4 4 4 4 4 4 4
•  4 4 4 4 4 4 4
•  4 4 4 4 4 4 4

For order n=5 there are 64 different arrangements with which 96 different symmetrical squares can be created. 2304 arrangements already exist for n=7 and the number of possible squares grows rapidly with increasing order.