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.
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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.

APPENDIX (this section can be skipped)

The Number of Group of Squares Generated for both Wheel Methods

The Number using the L-leap Boundary Method

The number of magic squares in each group may be determined by the use of two equations. The equation for the number of total groups, magic and non-magic De La Loubère using the L-leap approach uses the modified equation ½(n² + 1) - ½(n_s²- 1) - 1 which after simplification becomes ½(n² - n_s²), which is identical to Part I of De La Loubère method and gives similar results. The group consists of the internal wheel square along with the external L-leap boundary, i.e., making up the entire larger square. Although the internal square remains the same, the boundary may differ taking on a variety of values.

The Number using the Wheel Boundary Method

The total number of groups of magic modified wheel squares using the wheel approach to finish off the square uses the equation n_s(n - 1), same as in Part I. Again n_s and n correspond to the smaller and the larger square, respectively. The tables also shown in Part I for the Loubère method are identical to those used by Method C.