NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part Q2

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), i.e., ½(n2-n+2) to ½ n2+n, but may be chosen from any other consecutive group of numbers. However, the spoke on the consisting of the central column is no longer the adjacent numbers (22,23,24,25,26,27,28), where the number 23 correponding to ½(n2-3) is retained in (23,25,27) the next number on the list is ½(n2-5), i.e., 22, so that now the central column is (..,22,23,25,27,28,..) and (24,26) no longer belongs on the "spoke" but are replaced by a pair of complementary numbers (the ..) (see Square A2 below). Furthermore, the rest of the numbers belonging in the square are generated from the list below:


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 − 19 = 6.

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 {15,16,17,25,33,34,35} and {21,20,19,25,31,30,29}

  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=6, 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 24 is no longer on the bottom center column but is replaced by 18.
  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 99 which is not divisible by 2. The sum of pairs needed to fill up that line is obtained by investigating sums which may be possible. In this case (37 + 62) is a possible set. 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 101.
  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, (36,49), (38,26) 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, (8,13), (5,2) and (3,4) 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=6)
 
 
17 27 31
39 25 11
19 23 33
 
 
A2
15 32 2999
16 28 30 51
17 27 31
41 4039 25 11 109
19 23 33
20 22 34 49
21 18 35 101
98 49 51 102
A3
15 36 38 32 241 29
16 7 2844 30
17 27 31
41 4039 25 11 109
19 23 33
20 43 22 6 34
21 14 12 18 2649 35
A4
15 36 38 32 241 29
8 16 7 2844 30 42
5 3 17 2731 47 45
41 4039 25 11 109
4846 19 23 33 4 2
3720 43 22 6 34 13
21 14 12 18 2649 35
A5
15 3638 32 241 29
8 16 7 2844 30 42
5 3 17 2731 47 45
41 4039 25 11 109
484619 23 33 4 2
3720 43 22 6 34 13
21 14 12 18 2649 35
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 Q2 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part Q3 of an odd 9x9 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com