NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Part R2

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, which in our case may be (17,18,19,25,31,32,33) (the previous page) or (18,19,20,25,30,31,32) (this page).

Each will be treated separately since 4n + 3, where in this case n = 1 may be filled with all but one complementary set below:


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

or with all the complementary numbers as was shown previously. In the 4n + 1 numbers, however, both squares use all the numbers in their complementary set.

In addition, 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 (21,25,29) used to generate the left diagonal, 25 − 21 = 4.

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 {14,15,16,25,34,35,36} and {23,22,21,25,29,28,27}

  1. To the center column of the internal 3x3 square fill numbers 20,25,30 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. Thus the numbers follow the format of Square A1.
  2. With 21, 29 generate a 3x3 square using Δ=4, b=25 and a=30 so that the sum of each column, row and diagonal of the 3x3 square sums up to 75. (Square A1).
  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.
  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 102. The sum of pairs needed to fill up that line is obtained by investigating sums which may be possible. In this case (57 + 45) 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, (24,17), (8,13) 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, (2,3), (0,1) and (4,5) and enter into Square A4.
  11. Complementary set (9,41) cannot be used since a magic square is not possible.
  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=4)
 
 
16 30 29
38 25 12
21 20 34
 
 
A2
14 32 27102
15 31 28 51
16 30 29
40 3938 25 12 1110
21 20 34
22 19 35 49
23 18 36 98
98 49 51 102
A3
14 24 8 32 3733 27
15 7 3144 28
16 30 29
40 3938 25 12 1110
21 20 34
22 43 19 6 35
23 26 42 18 1317 36
A4
14 24 8 32 3733 27
2 15 7 3144 28 48
0 4 16 3029 46 50
40 3938 25 12 1110
4945 21 20 34 5 1
4722 43 19 6 35 3
23 26 42 18 1317 36
A5
14 248 32 3733 27
2 15 7 3144 28 48
0 4 16 3029 46 50
40 3938 25 12 1110
494521 20 34 5 1
4722 43 19 6 35 3
23 26 42 18 1317 36
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 R2 of a 7x7 Magic Square Wheel Spoke Shift method. To go to Part R3 of an odd 9x9 square.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com