NEW MAGIC SQUARES WHEEL METHOD

Part VIII

Picture of a wheel

7x7 Wheel Border Magic Square

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).

A second modified facile method for the construction of wheel type magic squares is now available. The position of the spokes are rotated by 90° so that the left diagonal starts at the bottom left cell. The 5x5 square is first filled followed by the 7x7. The 7x7 and the 5x5 squares are magic but not the 3x3 and this square is classified as a partial border.

The new magic squares with n = 7 are constructed as follows using a complimentary table as a guide.


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

A 7x7 Magic Square Using the Pairs {22,23,24,25,26,27,28} and {2,13,20,25,30,37,48}

  1. The 7x7 square is to be filled with 25 numbers from a non sequential subset of numbers and their complements. The spokes of the wheel are generated as follows: Numbers 22-28 in the left diagonal; numbers 2,13,20 and conjugates 48,37,30 in the right diagonal; numbers 1,12,21 and conjugates 49,38,29 in top to bottom center; and 3,14,19 and conjugates 47,36,31 in center horizontal (square A1). The addition of these pair of numbers and conjugates to the 7x7 square are shown below using directional pointed arrows:

    1 12 21 21320 31419 ... 222324
    25
    4938 29 483730 47 44 41 ... 263631
    ↓ ↖ → ... ↗
  2. Sum up the rows and columns 1-3 and 5-7 and subtract from the magic sum 175. This gives the amounts required (shown in green Square A2) The last column shows the two amounts need to complete the row and column (shown in green).
  3. Fill in the internal square 5x5 with the numbers 15-18 and complements 35-32 according to the picture below using two adjacent pair of numbers.
    4. Picture of arrows
  4. Then fill in the external non-spoke cells (rows 1 and 7 and columns 1 and 7) with the numbers 4 to 11 and complements 46 to 39 as in the picture below.
    5. Picture of arrows
A1
48 1 28
37 12 27
30 21 26
3 1419 25 31 3647
24 29 20
23 38 13
22 49 2
⇒
A2
48 1 28 98 49x2
37 12 27 99 49+50
30 21 26 98 48+50
3 1419 25 31 3647
24 29 20 102 50+52
23 38 13 101 50+51
22 49 2 102 51x2
102101102 9899 98
⇒
A3
48 1 28
37 17 1232 27
33 30 2126 15
3 1419 25 31 3647
18 24 29 20 34
23 35 38 16 13
22 49 2
⇒
A4
48 4 6 1 4345 28
42 37 17 1232 27 8
40 33 30 2126 15 10
3 1419 25 31 3647
1118 24 29 20 34 39
923 35 38 16 13 41
22 46 44 49 75 2
⇒
A4 Border
48 4 6 1 4345 28
42 37 17 1232 27 8
40 33 30 21 26 15 10
3 1419 25 31 3647
1118 24 29 20 34 39
923 35 38 16 13 41
22 46 44 49 75 2
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 VI of a border Magic Square Wheel method. To see the next 9x9 Part IX.
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Copyright © 2015 by Eddie N Gutierrez.