NEW MAGIC SQUARES WHEEL METHOD

Part N2

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). At least one pair of complements must be retained in the central column which may be (24,25,26) or (23,24,25,26,27) with the remaining two (22,28) or (22,23,27,28) replaced by respectively by 1 or 2 other complementary pairs 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, 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 {17,18,19,25,31,32,33} 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 24 → 25 → 26 starting from the 5th row (Square A1).
  2. With 19, 20 and their complements generate a 3x3 square using Δ=5, b=25 and a=26 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). 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 51.
  6. Repeat for rows 6 and 7 obtaining, respectively, 49 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, (9,11), (8,10) and (7,6) 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, (0,1), (2,3) and (4,5) and enter into Square A4.
  11. Since this is a 4n + 3 square at least one number is to be thrown out which in this case is 15.
  12. Figure A shows the connectivity between numbers in the complementary table where the red and blue bars are the "spoke" numbers.
  13. Picture of squares
    Figure A
  14. Square A5 shows the 3 border squares in "border format".
  15. The complement table below also shows how the color pairs are layed out (for comparison with Square A4).
A1 (Δ=5)
 
 
19 26 30
36 25 14
20 24 31
 
 
A2
17 34 2896
18 27 29 51
19 26 30
38 3736 25 14 1312
20 24 31
21 23 32 49
22 16 33 104
98 49 51 102
A3
17 9 8 34 4039 28
18 7 2744 29
19 26 30
38 3736 25 14 1312
20 24 31
21 43 23 6 32
22 41 42 16 1011 33
A4
17 9 8 34 4039 28
0 18 7 2744 29 50
2 4 19 2630 46 48
38 3736 25 14 1312
4745 20 24 31 5 3
4921 43 23 6 32 1
22 41 42 16 1011 33
A5
17 98 34 4039 28
0 18 7 2744 29 50
2 4 19 2630 46 48
38 3736 25 14 1312
474520 24 31 5 3
4921 43 23 6 32 1
22 41 42 16 1011 33
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 N2 of a 7x7 Magic Square Wheel Spoke Shift method. To see the next 7x7 Part N3.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com