NEW MAGIC SQUARES WHEEL METHOD

BORDER and SPOKE SHIFT Part A1

Picture of a wheel

How to Spoke Shift 7x7 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 left diagonal numbers don't have to be chosen from the consecutive group ½(n2-n+2) to ½(n2+n) but may be chosen from any other consecutive group of numbers. However, every spoke on the wheel consists of consecutive numbers and their complements. The example on this page, for comparisons sake, is based on the original complementary numbers (22,23,24,25,26,27,28) from the list:


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
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

based on ½(n2-n+2) to ½(n2+n) for the main diagonal. In addition, the square will be rotated 90° to the left to allow the lowest numbers to occupy the top of the central column. The symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first internal 3x3 magic square. Thus to obtain Δ for the regular wheel squares perform the following:

Δ = ½(n2 +1) − ½(n2-n+2), = ½(n − 1)

This formula is only to be used when the main right diagonal (left corner to right corner) consists of the numbers ½(n2-n+2) to ½(n2+n). When the sequence of numbers do not employ this set then Δ is calculated differently. (for example see the next page 7x7 Part A2).

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

A 7x7 Non-Transposed Magic Square Using the Diagonals {44,45,46,25,4,5,6} and {24,23,22,25,28,27,26}

  1. Generate a 3x3 square using Δ=3, b=25 and a=1. (Square A1).
  2. Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row (the "spoke") and include their complements from the complement list above.
  3. To begin filling up the square add up the entries on the first row and subtract from 175 (the magic sum for a 7x7 square). This affords the value 102 which divided by 2 gives the sum of pairs needed to fill up that line, i.e., 51 x 2.
  4. Repeat for row 2 except subtract the value from 125 (the magic sum for a 5x5 square). This gives a value of 51.
  5. Repeat for rows 6 and 7 obtaining, respectively, 51 and 102.
  6. Then repeat for columns 1 and 2 obtaining, respectively, 102 and 51.
  7. Finally repeat for columns 6 and 7 obtaining, respectively, 51 and 102.
  8. Repeat for columns 1 and 2 obtaining, respectively, 102 and 51 again.
  9. Fill the 2nd & 6th and 1th & 7th rows with the pairs/complements from the complement list corresponding to the requisite sums, (11,10) & (13,12), (15,14) and enter into Square A3.
  10. Fill the 2nd & 6th and 1st & 7th columns with the pairs/complements from the complement list corresponding to the requisite sums, (21,20) & (17,16),(19,18) and enter into Square A4.
  11. Square A5 shows the 3 border squares in "border format".
  12. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=3)
 
 
46 1 28
7 25 43
22 49 4
 
 
A2
44 3 26102
45 2 27 51
46 1 28
9 87 25 43 4241
22 49 4
23 48 5 51
24 47 6 102
102 51 51 102
A3
44 13 15 3 3638 26
45 11 240 27
46 1 28
9 87 25 43 4241
22 49 4
23 39 48 10 5
24 37 35 47 1412 6
A4
44 13 15 3 3638 26
17 45 11 240 27 33
19 21 46 128 29 31
9 87 25 43 4241
3230 22 49 4 20 18
3423 39 48 10 5 16
24 37 35 47 1412 6
A5
44 1315 3 3638 26
17 45 11 240 27 33
19 21 46 128 29 31
9 87 25 43 4241
323022 49 4 20 18
3423 39 48 10 5 16
24 37 35 47 1412 6
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
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26

This completes Part A1 of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 7x7 Part A2.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com