NEW MAGIC SQUARES WHEEL METHOD

Part I

7x7 Wheel Partially 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 square constructed has an internal 3x3 square which is not magic but whose 5x5 and 7x7 is. In addition the partially bordered square may be everted to give an opposite square whose internal 3x3 square is magic along with the external 7x7 square but not the internal 5x5.

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

A 7x7 Magic Square Using the Pairs {22,23,24,25,26,27,28} and {4,5,6,25,44,45,46}

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 4,5,6 and conjugates 44,45,46 in the right diagonal; numbers 1,2,3 and conjugates 47,48,49 in top to bottom center; and 7,8,9 and conjugates 41,42,43 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	2	3	4	5	6	7	8	9	...	22	23	24
   													25
   49	48	47	46	45	44	43	42	41	...	26	27	28
   ↓			↖			→			...	↗

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 external non-spoke cells (rows 1 and 7 and columns 1 and 7) with the numbers 10 to 17 and complements 33 to 40 as shown in Square A3 using adjacent pair of numbers to fill the row and columns according to A or B in the picture:

4. Fill in the internal square 5x5 with the numbers 18-21 and complements 29-32 according to C above using two adjacent pair of numbers. This ensures that the square is magic.

Conversion of the 7x7 into its transposed opposite

Generation of a 7x7 transposed opposite can also follow the route used above. The method to transpose columns followed by rows follows. This generates a new square in which the internal 3x3 and the external 7x7 are magic.

1. Take square A4 and transpose (column 1 with column 3) and (column 5 with column 7) to get Square A5.
2. Take square A6 and transpose (row 1 and row 3) and (row 5 with row 7) to get Square A6.
3. In a sense A4 has been imploded or everted into A6, i.e., A4 and A6 below are opposites.

The result is a new everted square conforming to the same complementary table.

This completes part I of a border Magic Square Wheel method. To see the next 7x7 Part II.
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