New Squares from the Modified Wheel Boundary Method

Part IIIC

The wheel approach:Method C Continued

For these squares the triplet is added to square first, followed a second triplet. The only difference is that two of the triplet pairs must be added in an opposite manner. This is similar to Method B but only for this size square. Larger n follow the Method C pattern.

For example, for a 5x5 square II (below), the two triplets {1,2,3} and {4,5,6} along with their complements {25,24,23} and {22,21,20} are taken from the complementary table below and used to construct the "spokes".

This example being a variant of Example A, is constructed as follows:

1. Incorporate the smaller square into a larger 5x5 square as shown below.
2. Fill in the spoke cells, note that the 3 and 1 are reversed from Method C: Example A.
3. Fill in the non-spoke cells as shown in method A variant 1.

Note that with this method using a 3x3 expanded into a 5x5 the same or similar results are obtained as in method B. However, when nxn is greater than 3x3 the results are not the same, and thus the reason for naming this method C.

The following example A shows the construction of a 7x7 by adding a triplet first followed by the triad {(1,2),(3,4),5,6)} using the wheel method

1. Fill in the main diagonal with consecutive numbers 22-28.
2. Fill in the triplet spoke cells for the 3x3.
3. Fill in the outer 7x7 with the triad spoke cells (note that 11 and 13 are reversed from Example A).
4. Fill in the internal blue color non-spoke cells.
5. Fill in the outer rest of the non-spoke cells as shown in method A variant 1 first top/bottom rows then side left/right columns.
6. Compare to this Example B to Method C: Example A.

The following table summarizes the pairs of numbers and their parities required to fill in the empty rows or columns (spoke numbers) for last square 5 above, where two numbers e.g., 36/15 or 32/18 correspond to a pair.

This above example (7x7)is different from method B in that six pairs of consecutive numbers are used to create a square, followed by 3 pairs of numbers to complete the square. Method B uses 3 separate pairs three times to construct the 7x7 square.

This completes this section on new squares from modified wheel methods. The next page will show a new method D (Part IV) using spoke as well as anti-spoke additions to produce new magic squares. To return to previous page (Part IIIA) or to return to homepage.