NEW MAGIC SQUARES WHEEL METHOD - BORDER SQUARES

Part X2

How to generate 9x9 Border Magic Squares

A magic square is an arrangement of numbers 1,2,3,... n² where every row, column and diagonal add up to the same magic sum S and n is also the order of the square. A magic square having all pairs of cells diametrically equidistant from the center of the square and equal to the sum of the first and last terms of the series n² + 1 is also called associated or symmetric. In addition, the center of this type of square must always contain the middle number of the series, i.e., ½(n² + 1).

This site introduces a new method used for the construction of border wheel type squares. The method consists of forming a 5x5 internal Wheel spoke magic square then filling in the external 1,2 and 8,9 rows and columns with the requisite non-spoke numbers as will be shown below.

Furthermore, the symbol δ (where δ = 4) specifies the difference between entries on the diagonals and center row and column not situated on the center 5x5 square (shown in square A1).

The non-wheel entries in rows and columns 1,2 and 8,9 are added according to a coded system ( which I call "coded connectivity" as opposed to lined connectivity) employing a number and superscript and where the number gives the difference between two paired numbers and the superscript shows which two numbers are paired together. For example, 11¹ says that this number is added to a second complementary number 11¹ separated by a distance of 11. From the complementary table above 1 + 71 is such an example. While, 7^a means that this number is added to a non-complementary number 7^a both which are 7 units apart. In addition, if we look at the complementary table above 2¹ corresponds to the sum of 1 + 80, while 2^-1 to the sum of 2 + 81. When either of the two sums is required, the number is preceded by either a ( ) or by (-).

A 9x9 Transposed Magic Square Using the Diagonals {60,56,51,50,41,32,31,26,22} and {24,28,39,40,41,42,43,54,58}

1. Fill in the 5x5 internal square with the 25 numbers from the subset 29 to 53 (square A1). The spokes of the wheel are generated as follows: Numbers 39-43 in the left diagonal; numbers 31,32 and conjugates 50,51 in the right diagonal; numbers 29,30 and conjugates 52,53 in top to bottom center; and 33,34 and conjugates 48,49 in center horizontal. The addition of these pair of numbers and conjugates to the 5x5 square are shown below using directional pointed arrows:

   29	30	31	32	33	34	35	36	37	38	39	40
   													41
   53	52	51	50	49	48	47	46	45	44	43	42
   ↓		↖		→						↗

2. The 5x5 square is finished by filling in the non-spoke numbers from those numbers left over, i.e., 35 to 38 and their conjugates 44 to 47, giving a magic sum of 205.
3. Subtract δ = 4 from 29 and add this number (25) to the center cell of row 2 and repeat again (25-4) = 21 and place this number in the center cell of row 1. Starting ith 21 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 28, followed by their complementary numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 5x5 square.
4. Sum up the empty 1^st row, the empty 9^th; the 2^nd, the 8^th rows. Do the same for the columns (green cells). See Square A2.
5. Fill in the internal 7x7 square with numbers generated using the new coding method (Square 3). For example in row 2 using numeric superscripts, 2 is added to 74 and 4 to 72 , followed by their complements. While in column 2 also using numeric superscripts, 11 is added to 77 and 18 to 70 also followed by their complements.
6. Finally fill in the external 9x9 square (color cells) (Square A4). For example, in row 1, 3 is added to 75, 13 to 65 and 1 to 73. While in column 9, 67 is added to 19, 66 to 20 and 76 to 14 followed by their complements, using numeric superscripts.
7. Below is the coded connections to this square where the colored "spoke" cells are not included in the coding:

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	...	40
																						41
81	80	79	78	77	76	75	74	73	72	71	70	69	68	67	66	65	64	63	62	...	42
91	71	51	72	73	92	51	71	91	72	73	74	52	92	53	54	52	74	53	54	...

8. Because of the messy connectivities using spaghetti lines its best to use the connectivity table and the table at the end of this page to assertain the connectivities.
9. Square A5 shows the 3 border squares in "border format".
10. The complementary table below also shows how the color pairs are layed out (for comparison with Square A4).
11. This square differs from the magic square A5 generated in the previous page in that only the 5x5 internal squares differ. The entries in rows and columns 1,2,8,9 remain the same.

This completes Part X2 of a 9x9 Magic Square Wheel Spoke Shift method. To go to Part X3 of an 11x11 square.
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