NEW MAGIC SQUARES WHEEL METHOD

Part III

Picture of a wheel

9x9 Magic Square Wheel

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).

A modified facile method for the construction of wheel type magic squares is now available. The position of the spokes are rotated by 90° so that the left diagonal starts at the bottom left cell. The 9x9 square constructed has no internal border squares. In addition the partially bordered square may be everted to give an opposite square whose internal 7x7 square is not magic but whose 3x3, 5x5 and 9x9 are magic.

The new magic squares with n = 9 are constructed as follows using a complimentary table as a guide.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62
 
21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40
41
61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

A 9x9 Transposed Magic Square Using the Diagonals {37,38,39,40,41,42,43,44,45} and {5,6,7,8,41,74,75,76,77}

  1. The 9x9 square is to be filled with 33 numbers from the subset 1-12 and their complements 70-81 and the numbers 37-45. The spokes of the wheel are generated as follows: Numbers 37-45 in the left diagonal; numbers 5,6,7,8 and conjugates 74,75,76,77 in the right diagonal; numbers 1,2,3,4 and conjugates 76,79,80,81 in top to bottom center; and 9,10,11,12 and conjugates 70,71,72,73 in center horizontal (square A1). The addition of these pair of numbers and conjugates to the 9x9 square are shown below using directional pointed arrows:

    1 2 3 4 5 6 789 101112 ... 373839 40
    41
    8180 79 787776 75 74 73 72 71 70 ... 454443 42
    ...
  2. Sum up the rows and columns 1-4 and 6-9 and subtract from the magic sum 369. This gives the amounts required (shown in green Square A2). The last column shows the two amounts need to complete the row and column (shown in yellow).
  3. Fill in the non-spoke cells of the internal 5x5 square with the numbers 21 to 24 and complements 58 to 61 as shown in Square A3 using four adjacent pair of numbers according to inset C in the picture:
  4. Picture of arrows
  5. Fill in the non-spoke cells of the outer rows of the internal square 7x7 with the numbers 13-24 and complements 62-69 according to inset A or B above using two adjacent pair of numbers.
  6. Fill in the non-spoke cells of the outer rows of the external square 9x9 with the numbers 25-36 and complements 46-57 according to inset A above using two adjacent pair of numbers.
  7. Note that the non-spoke numbers of the entire 7x7 square is laid out according to the method of Part I and is, therefore, easy to construct. The outer rows and columns of the 9x9 square then just require either two numbers summing up to 81 or 83.
A1
77 1 45
  76 2 44
  75 3 43
  74 4 42
910 11 12 41 70 717273
40 78 8
  39 79 7
 38 80 6
37 81 5
A2
77 1 45246 82x3
  76 2 44 24783x2+81
  75 3 43 248 83x2+82
  74 4 42 24982x2+84
910 11 12 41 70 717273
40 78 8 24382x2+80
  39 79 7 24481x2+82
 38 80 6 24581x2+83
37 81 5 24682x3
246245244 243249 248247246
A3
77 1 45
  76 2 44
7522 3 61 43
60 74 4 42 24
910 11 12 41 70 717273
2140 78 8 59
39 58 79 23 7
 38 80 6
37 81 5
A4
77 1 45
76 13 16 267 68 44
647522 3 61 43 18
6360 74 4 42 24 19
910 11 12 41 70 717273
202140 78 8 5962
1739 58 79 23 7 65
38 69 66 80 1514 6
37 81 5
A5
77 25 2729 1 5355 57 45
51 76 13 16267 68 44 32
4964 7522 3 61 43 18 34
4763 60 74 4 42 24 1935
910 11 12 41 70 717273
362021 40 78 8 59 6246
3317 39 58 79 23 7 6548
3138 69 66 80 1514 6 50
37 56 5452 81 3028 26 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
 
81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56
 
27 28 29 30 3132 33 34 35 36 37 38 39 40
41
55 54 53 52 51 50 49 48 47 46 45 44 43 42

Conversion of the 9x9 into its transposed opposite

Generation of a 9x9 transposed opposite can also follow the route used above. Unfortunately as n > 5 their generation becomes more and more complicated. A method that obviates this is to transpose columns followed by rows. This generates a new square which is not a border square. Only the external square is magic.

  1. Take square A5 and transpose (column 1 with column 4), (column 2 with column 3), (column 6 with column 9) and (column 7 with column 8) to get Square A6.
  2. Take square A6 and transpose (row 1 with row 4), (row 2 with row 3), (row 6 with row 9) and (row 7 with row 8) to get Square A7.
  3. In a sense A5 has been imploded or everted into A7, i.e., A5 and A7 below are opposites.
A5
77 25 2729 1 5355 57 45
51 76 13 16267 68 44 32
4964 7522 3 61 43 18 34
4763 60 74 4 42 24 1935
910 11 12 41 70 717273
362021 40 78 8 59 6246
3317 39 58 79 23 7 6548
3138 69 66 80 1514 6 50
37 56 5452 81 3028 26 5
A6
29 27 2577 1 4557 55 53
16 13 76 51 2 32 44 68 67
2275 6449 3 34 18 43 61
7460 63 47 4 35 19 2442
1211 10 9 41 73 727170
402120 36 78 46 62 598
5839 17 33 79 48 65 723
6669 38 31 80 506 14 15
52 54 5637 81 5 2628 30
A7
7460 63 47 4 35 19 2442
2275 6449 3 34 18 43 61
16 13 76 51232 44 68 67
29 27 2577 1 4557 55 53
1211 10 9 41 73 727170
52 54 5637 81 5 26 28 30
6669 38 31 80 506 14 15
5839 17 33 79 48 65 723
402120 36 78 46 62 598
A7 Partially Bordered
7460 63 47 4 35 19 2442
2275 6449 3 34 18 43 61
16 13 76 51232 44 68 67
29 27 2577 1 4557 55 53
1211 10 9 41 73 727170
52 54 5637 81 5 26 28 30
6669 38 31 80 506 14 15
5839 17 33 79 48 65 723
402120 36 78 46 62 598

The result is a new square conforming to the same complementary table above and where the 3x3, 5x5 and 9x9 are magic but the 7x7 border square is not.

This completes Part III of a 9x9 border Magic Square Wheel method. To go to Part IV of an 9x9 square.
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Copyright © 2015 by Eddie N Gutierrez