NEW MAGIC SQUARES WHEEL METHOD

Part P2

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, the spoke of the central column is no longer the adjacent numbers (22,23,24,25,26,27,28). The only numbers retained are (23,25,27) with the remaining two complements (22,28) and (24,26) being replaced by another complementary pair 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

numbers which again 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 (20,25,30) used to generate the left diagonal, 25 − 20 = 5.

In addition, both 4n + 1 and 4n + 3 squares may be filled with the entire complement set. Previously the 4n + 3 square had to be filled with at least one 0 or negative number. See the 7x7 square.

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 {16,17,18,25,32,33,34} and {22,21,20,25,30,29,28}

  1. To the center column of the internal 3x3 square fill numbers ½(n2-1) to ½(n2+3) in consecutive order starting at the bottom middle cell of the 3x3 internal square and proceeding to the top middle cell of the 3x3 internal square using the numbers listed in the complementary table described above, as for example using n = 7. For a 7x7 square the numbers in the center column correspond to 23 → 25 → 27 starting from the 5th row (Square A1).
  2. With 18, 20 and their complements generate a 3x3 square using Δ=5, b=25 and a=27 so that the sum of each column, row and diagonal of the 3x3 square sums up to 75. (Square A2).
  3. 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. Note that 22 is no longer on the bottom center column but is replaced by 16.
  4. 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 96 which divided by 2 gives the sum of pairs needed to fill up that line, which in this case may be (48 x 2) but in this case is (35 + 61). See Figure A2.
  5. Repeat for row 2 except subtract the value from 125 (the magic sum for a 5x5 square). This gives a value of 48.
  6. Repeat for rows 6 and 7 obtaining, respectively, 52 and 104.
  7. Then repeat for columns 1 and 2 obtaining, respectively, 98 and 49.
  8. Finally repeat for columns 6 and 7 obtaining, respectively, 51 and 102.
  9. Fill the 1st & 7th and 2nd & 6th rows with the pairs/complements from the complement list corresponding to the requisite pairs, (9,24), (14,3) and (8,10) and enter into Square A3.
  10. Fill the 1st & 7th and 2nd & 6th columns with the pairs/complements from the complement list corresponding to the requisite pairs, (4,5), (1,2) and (6,7) and enter into Square A4.
  11. Figure A shows the connectivity between numbers in the complementary table where the red and blue bars are the "spoke" numbers.
  12. Picture of squares
    Figure A
  13. Square A5 shows the 3 border squares in "border format".
  14. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=5)
 
 
18 27 30
37 25 13
20 23 32
 
 
A2
16 35 2896
17 31 29 48
18 27 30
39 3837 25 13 1211
20 23 32
21 19 33 52
22 15 34 104
98 49 51 102
A3
16 9 14 35 4726 28
17 8 3140 29
18 27 30
39 3837 25 13 1211
20 23 32
21 42 19 10 33
22 41 36 15 324 34
A4
16 9 14 35 4726 28
4 17 8 3140 29 46
1 6 18 2730 44 49
39 3837 25 13 1211
4843 20 23 32 7 2
4521 42 19 10 33 5
22 41 36 15 324 34
A5
16 914 35 4726 28
4 17 8 3140 29 46
1 6 18 2730 44 49
39 3837 25 13 1211
484320 23 32 7 2
4521 42 19 10 33 5
22 41 36 15 324 34
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 P2 of a 9x9 Magic Square Wheel Spoke Shift method. To see the next 9x9 Part P3.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com