NEW MAGIC SQUARES WHEEL METHOD - SPOKE SHIFT

Picture of a wheel

How to Spoke Shift 3x3 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 two new methods used for the construction of wheel type squares except that the initial spoke parts are added in a somewhat different manner than in the original wheel method. The first method consists of pairing numbers in complementary fashion, partitioning these complementary pairs into groups, generating in the spoke and then filling in the non spoke cells with the remaining complementary pairs as was done in the original method. The difference between this type of square and the original is that the numbers 0, -1,-2 etc and their complements may now part of the square. Since the number of cells in an nxn magic squares is n then a complementary pair (not containing 0 or any minus number) is not used in generating the square. The use of these other numbers is a requirement because the use of numbers from 1 to n may not be enough to generate this type of magic square.

In addition, the diagonal pairs are obtained from the complementary table using what I call a "Cross-Over" method shown below. For a square with n = 3, there are two sets of pairs. These are the (a) {1} and {2}, and (b) {2} and {3}. These pairs and their complements make up entries to the diagonal cells. The first square having the diagonal {1} and {2} and the second having the diagonal {2} and {3} are discussed below. The first is an odd/even diagonal the second an even/odd one. Both together give two possible squares for n = 3. A way of predicting the total number of Cross-Overs is shown in Table Fa and Table Fb.

A 3x3 Magic Square Using the Pairs {1} and {2}

  1. The center column is filled with the group of numbers ½ (n2-n+2) to ½(n2+n) in consecutive order starting at the bottom cell and proceeding to the top cell from the numbers listed in the complementary table described above, for example using n = 5. For a 3x3 square the numbers in the center column correspond to 4 → 5 → 6 starting from the bottom (Square A1).
  2. This is followed by adding the -2 to the center row with -2 to the right of 5, and then adding its complement (12) to the left of 5 (Square A2).
  3. 2 pairs are left with which to construct the spoke and fill in the non-spoke cells. To obtain the spoke cells we take the pairs {1} and {2} where the midpoint between 1 and 2 serves as a "Cross-Over" between these pair of numbers i.e. (1) and (2). The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A3. The numbers 1 and its complement 9 are added down to the right and 2 and its complement 8 are added up right as shown.
  4. The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 4 → 5 → 6 have been transposed or shifted to a column.
  5. The square that is produced via this method (A3) is magic since the sum of all rows, columns and diagonals are 39.
  6. In addition, the complementary table shows that the numbers -1,0,3 and their complements are not used.
A1
6
5
4
A2
6
12 5 -2
4
A3
1 6 8
12 5 -2
2 4 9
-2 -1 0 1 2 3 4
5
12 11 10 9 8 7 6

A 3x3 Magic Square Using the Pairs {2} and {3}

  1. The center column is filled with the group of numbers ½ (n2-n+2) to ½(n2+n) in consecutive order starting at the bottom cell and proceeding to the top cell from the numbers listed in the complementary table described above, for example using n = 5. For a 3x3 square the numbers in the center column correspond to 4 → 5 → 6 starting from the bottom (Square B1).
  2. This is followed by adding the 0 to the center row with 0 to the right of 5, and then adding its complement (10) to the left of 5 (Square B2).
  3. 2 pairs are left with which to construct the spoke and fill in the non-spoke cells. To obtain the spoke cells we take the pairs {2} and {3} where the midpoint between 2 and 3 serves as a "Cross-Over" between these pair of numbers i.e. (2) and (3). The first set (with complements) corresponds to the left diagonal and the second set to the right as shown in Square A3. The numbers 2 and its complement 8 are added down to the right and 3 and its complement 7 are added up right as shown.
  4. The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 4 → 5 → 6 have been transposed or shifted to a column.
  5. The square that is produced via this method (A3) is magic since the sum of all rows, columns and diagonals are 39.
  6. In addition, the complementary table shows that the number 1 and its complement is not used.
B1
6
5
4
B2
6
10 5 0
4
B3
2 6 7
10 5 0
3 4 8
0 1 2 3 4
5
10 9 8 7 6

This completes the 3x3 Magic Square Wheel Spoke Shift Part A method.
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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com