New Squares from the Modified Wheel Boundary Method

Part IIIA

A Discussion of the new Wheel type squares

the previous page showed how to prepare Modified Wheel squares of order n where the main diagonal is constructed from a complementary table of (n+2)x(n+2) or greater. this page will show how we can take these modified magic squares and build them up into larger squares that take into account all of the (n+2)x(n+2) or greater numbers of the complementary table used. Two methods will be shown one using a L-leap approach, the other a wheel approach.

the L-leap approach

To expand a modified wheel square using the L-leap approach (leaping across the square in a z manner) we take an nxn and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:

1. Incorporate the smaller square as constructed previously into a larger square as shown below using a 3x3 inserted into a 5x5.
2. Fill in the two corner cells to complete the main diagonal.
3. Fill in the first cell adjacent to the 15 in this example with the number 4, since 3 was the last number on the complementary list.
4. Perform a L-leap, i.e. place one number in a cell L-leaping across up/down on the square to the next column or right/left to the next row and insert the next number.
5. Stop at the fifth cell in the second column inserting a 6, leap across to the right upper corner and insert a 7, then leap to the fourth cell in the first column and insert an 8.
6. Continue L-leaping right and left across until you reach the first cell in the second row.
7. Take the complement of 10, i.e. 16, and this time L-leap right and left across the square, filling in all the complementary cells up to the left lower corner cell in the first column.
8. L-leap down and up filling in all the cells with the right complementary numbers until the square is completed.

The above gives one way of filling up a square by the L-leap approach. A second method fills it in an opposite manner gives a different variant:

1. Use square 2 from above.
2. Fill in the first up cell adjacent to the 15 in this example with the number 4.
3. Perform a L-leap, i.e. place one number in a cell L-leaping across the square left/right to the next row and insert the next number.
4. Stop at the second cell in the fifth column inserting a 6, leap across to the left lower corner and insert a 7, then leap to the first cell in the fourth column and insert an 8.
5. Continue L-leaping up and down until the first cell in the second column is reached.
6. Take the complement of 10, i.e. 16, and this time L-leap backwards(up/down) across the square, filling in all the complementary cells up to the top right corner cell in the fifth column.
7. L-leap left and right filling in all the cells with the right complementary numbers until the square is completed.

Method C produces two types of magic squares, depending on the whether the single triplet {a,b,c} is added at the end to the boundary of the nxn square, as for example in the 7x7 partial square 2-3 below (Example A) generated from the triad {(1,2),(3,4),5,6)} and complements or the triplet is added to square first, followed by the same triad. the latter approach while not really filling in the external boundary, however, is part of method C. the only difference is that two of the triplet pairs must be added in an opposite manner (see next page Example B).

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".

To expand a modified wheel square using the wheel approach we take an nxn square and increase the number of rows and columns by 2 or greater. Using the 3x3 example we:

1. Incorporate the smaller square as constructed in new Loubère methods into a larger square as shown below using a 3x3 inserted into a 5x5.
2. Fill in the spoke cells.
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 method C.
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The following example A shows the construction of a 5x5 internal partial square expanded into a 7x7 using the wheel method

1. Fill in the main diagonal with consecutive numbers 22-28.
2. Fill in the spoke cells for the 5x5.
3. Either complete the internal square or fill in the outer 7x7 spoke cells.
4. Fill in the internal 5x5 non-spoke cells.
5. Fill in the outer boundary non-spoke cells as shown in method A variant 1 first top/bottom rows then side left/right columns.

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.

The next page shows a continuation of method C con't.
To return to previous page (Part II) using 7x7 squares or to return to homepage.