NEW MAGIC SQUARES WHEEL METHOD - BORDER SQUARES

Part X3

Picture of a wheel

How to generate 11x11 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 methods used for the construction of border wheel type squares except that the initial spoke parts are added in a somewhat different manner than in the original wheel method. The method consists of forming an internal 3x3 magic square, then generating all subsequent border magic squares. as was done in the original method. The difference between this type of square and the original is that the numbers (and complements) are added consecutively, starting from 1, at the center top cell. Subsequent numbers are added to each of the diagonals and the center row. The left diagonal of the internal 3x4 square, however, deviates from this arrangement where the three numbers on this diagonal are ½(n2 − 1), ½(n2 + 1), ½(n2 + 3).

This site introduces a new method used for the construction of border wheel type squares. The method consists of forming a 7x7 internal Loubère magic square then filling in the external 1,2 and 10,11 rows and columns with the requisite non-Loubère numbers as will be shown below.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
 
101 100 99 98 97 96 95 94 93 92 91 90 89 48 87 86 85 84 83 82
 
41 42 43 44 45 46 47 48 49 50 5152 53 54 55 56 57 58 59 60
61
81 80 79 78 77 76 75 74 73 72 71 60 69 68 67 66 65 64 63 62

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 7x7 square (shown in square A1)

The non-Loubère entries in rows and columns 1,2 and 10,11 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, 7a means that this number is added to a non-complementary number 7a 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 11x11 Transposed Magic Square Using the Diagonals {32,36,58,59,60,61,62,63,64,86,90} and {30,34,56,39,78,61,44,83,66,88,92}

  1. Fill in the internal 7x7 square with the numbers 37 through 85 to form a magic square.
  2. Subtract δ = 4 from 37 and add this number (33) to the center cell of row 2 and repeat again (33-4) = 29 and place this number in the center cell of row 1. Starting with the number 29 place consecutive numbers as shown in Square A1 in a spiraling fashion up to the number 36, followed by their complementary numbers (Square A1). This completes the spokes for the square, along with the nonspoke numbers of the 7x7 square.
  3. Thus the spokes of the wheel are shown as follows: Left diagonal 32,36...86,90; right diagonal 30,34...88,92; central column 29,33...89,93; central row 31,35...87,91. (...) denotes the numbers from the 7x7 square (Square A1).
  4. Sum up the rows with the diagonal and central row or column and subtract from 549 (sum of 9x9 internal square), to give the amounts required to complete the 9x9 square. The 13th rows shows the numbers required. (Square A2).
  5. Fill in the required pairs for the row and columns chosen from the coded table below coded in numeric superscripts.
  6. Repeat for rows and columns 1,11. However, subtract the numbers from 671 (the sum of a 11x11 magic square). Square A2 shows 4 pairs of numbers are required and these are listed down. The coded table below shows which numbers pair up. Note that the sum of the numbers in rows and columns 1 and 11 sum up to 122 (Square A4).
  7. Below is the coded connections to this square where the colored "spoke" cells and those labeled (...) i.e., the 7x7 entries, are not included in the coding.
    8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    121 120 119118117116115114113 112 111110 109108107 106
    261 91 262 92 93 94 95 96 31 91 31 92 93 94 95 96
    17181920212223 2425 262728...58 59 60
    61
    10510410310210110099 9897 969594...64 63 62
    3233 32 33 2121 22 22 34261 34 262
  8. Square A5 shows the 3 border squares in "border format".
  9. The complementary table below also shows how the color pairs of Square A4 are layed out.
A1
92 29 90
  88 33 86
  66 75 8437 46 55 64
  74 83 4345 54 63 65
 82 42 44 53 62 71 73
313541 50 52 61 70 7281 8791
49 51 60 69 78 80 40
  5759 6877 79 39 48
  58 67 76 85 38 47 56
  36 89 34
32 93 30
⇒
A2
92 29 90 460 123+97+(120x2)
  88 33 86 342114x3
  66 75 8437 46 55 64
  74 83 4345 54 63 65
 82 42 44 53 62 71 73
313541 50 52 61 70 7281 8791
49 51 60 69 78 80 40
  5759 6877 79 39 48
  58 67 76 85 38 47 56
  36 89 34 390130x3
32 93 30 516 121+(121x2)+147
516390 342460
⇒
A3
92 29 90
88 2 45 33 109 110112 86
116 66 7584 3746 55 54 6
11574 8343 45 54 63 65 7
11482 42 44 53 62 71 738
313541 50 52 61 70 7281 8791
164951 60 69 78 80 40106
1557 59 68 77 79 39 48 107
14 58 67 76 85 3847 56 108
36 120 118117 89 1312 10 34
32 93 30
⇒
A4
9222 917 129 96103 111101 90
9888 2 45 33 109 110112 8624
104116 66 7584 3746 55 64 618
9711574 8343 45 54 63 65 725
11911482 42 44 53 62 71 738 3
313541 50 52 61 70 7281 8791
281649 5160 69 78 80 4010694
271557 59 68 77 79 39 48 10795
2014 58 67 76 85 3847 56 108 102
2336 120118 117 89 1312 10 34 99
32100 113105 12193 2619 1121 30
⇒
A5 Partial Border
9222 9 171 29 96 103 111 101 90
9888 2 45 33 109 110112 8624
104116 66 7584 3746 55 64 6 18
97115 74 8343 45 54 63 65 7 25
119114 824244 53 62 71 73 83
3135 41 50 52 61 70 728187 91
2816 4951 60 69 78 8040106 94
27 15 5759 68 77 7939 48107 95
2014 58 67 76 85 3847 56 108 102
2336 120 118117 89 13 12 10 3499
32 100 113 105 12193 26 19 1121 30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 41
 
101 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81
42 43 44 4546 4748 49 5051 5253 54 5556 57 5859 60
61
8079 7877 76 7574 73 7271 7069 68 6766 65 6463 62

This completes the Part X3 of a 11x11 Magic Square Wheel Spoke Shift method. To go to Part X4 of an 11x11 square.
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Copyright © 2015 by Eddie N Gutierrez