NEW MAGIC SQUARES WHEEL METHOD

Part VI

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 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 7x7 as well as its internal internal squares. In addition, eversion of the square gives an opposite square which is not bordered.

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 {2,5,8,25,42,45,48}

  1. The 7x7 square is to be filled with 25 numbers from the subset 1-9 and their complements 41-49 and the numbers 22-28. The spokes of the wheel are generated as follows: Numbers 22-28 in the left diagonal; numbers 2,5,8 and conjugates 48,45,42 in the right diagonal; numbers 1,4,7 and conjugates 49,46,43 in top to bottom center; and 3,6,9 and conjugates 47,44,41 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 4 7 258 369 ... 222324
    25
    4946 43 484542 47 44 41 ... 262728
    ↓ ↖ → ... ↗
  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 yellow).
  3. Fill in the internal square 5x5 with the numbers 10-13 and complements 37-40 according to the picture below using two adjacent pair of numbers.
  4. Then similarly fill in the external non-spoke cells (rows 1 and 7 and columns 1 and 7) with the numbers 14 to 21 and complements 29 to 36.
    5. Picture of arrows
A1
48 1 28
45 4 27
42 7 26
3 69 25 41 4447
24 43 8
23 46 5
22 49 2
⇒
A2
48 1 28 98 49x2
45 4 27 99 49+50
42 7 26 100 50+50
3 69 25 41 4447
24 43 8 100 50+50
23 46 5 101 50+51
22 49 2 102 51x2
102101100 10099 98
⇒
A3
48 1 28
45 10 439 27
38 42 726 12
3 69 25 41 4447
13 24 43 8 37
23 40 46 11 5
22 49 2
⇒
A4
48 14 16 1 3335 28
32 45 10 439 27 18
30 38 42 726 12 20
3 69 25 41 4447
2113 24 43 8 37 29
1923 40 46 11 5 31
22 36 34 49 1715 2
⇒
A4 Border
48 14 16 1 3335 28
32 45 10 439 27 18
30 38 42 7 26 12 20
3 69 25 41 4447
2113 24 43 8 37 29
1923 40 46 11 5 31
22 36 34 49 1715 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

A Second Magic Square of the Same Type

Many Magic squares of the wheel type can be generate by employing a different set of numbers for the spokes of the wheel. If we use numbers 13-21 and their complements 37-29 as shown below, square A5 is generated.

A5
36 5 7 13 4244 28
41 33 1 1648 27 9
39 47 30 1926 3 11
15 1821 25 29 3235
144 24 31 20 46 38
1023 49 34 2 17 40
22 45 43 37 86 14
... 13 16 19 141720 151821 222324
25
... 3734 31 363330 35 32 29 262728
... ↓ ↖ →↗

This completes part VI of a border Magic Square Wheel method. To see the next 9x9 Part VII.
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Copyright © 2015 by Eddie N Gutierrez.