NEW MAGIC SQUARES WHEEL METHOD

Part M4

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. For example, the 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

are placed in the central column and not on the diagonal. 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. In this case Δ is, consequently, obtained from the first number of the set (15,16,17) used to generate the left diagonal, 25 − 15.

In addition, 4n + 1 number behave differently from 4n + 3 in that in the former at least one number in the set must be less than or equal to 0, while in the latter all numbers from 1 to ½(n2 − 1) are usable half of the time.

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

A 7x7 Transposed Magic Square Using the Diagonals {12,13,14,25,36,37,38} and {18,17,16,25,34,33,32}

  1. Generate a 3x3 square using Δ=10, b=25 and a=26. (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, which in this case may be (51 x 2) but because of the way the complement table is structured (in groups of three) is (47 + 55). See Figure A2.
  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, 49 and 98.
  6. Then repeat for columns 1 and 2 obtaining, respectively, 102 and 51.
  7. Finally repeat for columns 6 and 7 obtaining, respectively, 49 and 98.
  8. Fill the 1st & 7th and 2nd & 6th rows with the pairs/complements from the complement list corresponding to the requisite sums, (0,3), (7,2) and (19,18) and enter into Square A3.
  9. Fill the 1st & 7th and 2nd & 6th columns with the pairs/complements from the complement list corresponding to the requisite sums, (9,8), (11,10) and (21,20) and enter into Square A4.
  10. Figure A shows the connectivity between numbers in the complementary table where the red bars are the "spoke" numbers.
  11. Picture of squares
    Figure A
  12. Square A5 shows the 3 border squares in "border format".
  13. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=6)
 
 
14 26 35
46 25 4
15 24 36
 
 
A2
12 28 33102
13 27 34 51
14 26 35
44 4546 25 4 56
15 24 36
16 23 37 49
17 22 38 98
102 51 49 98
A3
12 0 7 28 4847 33
13 19 2732 34
14 26 35
44 4546 25 4 56
15 24 36
16 31 23 18 37
17 50 43 22 23 38
A4
12 0 7 28 4847 33
9 13 19 2732 34 41
11 21 14 2635 29 39
44 4546 25 4 56
4030 15 24 36 20 10
4216 31 23 18 37 8
17 50 43 22 23 38
A5
12 07 28 4847 33
9 13 19 2732 34 41
11 21 14 2635 29 39
44 4546 25 4 56
403015 24 36 20 10
4216 31 23 18 37 8
17 50 43 22 23 38
0 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
50 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 M3 of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 9x9 Part M5.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com