New Squares from the Modified De La Loubère Method Con't

Part IC

A Discussion of New De La Loubère type squares: the cross L-leap algorithm

The previous page showed a modified De La Loubère squares of order 3. This page is a continuation in that the L-leap method is modified so that the insertion of certain numbers into the boundary cells must go through a variable arm length cross path. The cross is normal only for a 5x5 and is distorted or unsymmetrical for 7x7 or larger.

Modified Method IV using the cross L-leap approach

To expand a modified De La Loubère square using the cross L-leap approach :

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 5x5 diagonal and fill in the first cell in the second row with the number 4, since 3 was the last number on the complementary list.
3. Perform an L-leap, to the right lower corner insert a 5 and proceed up and across the square to the number 8, the first arm of the cross. Move to just below and to the left of the number 4 (the number 1, the center of the cross) and move up one cell and insert a 9, the second arm of the cross.
4. Move back away from the 4 in an L-leap way (since moving towards the 4 inserts the wrong complement for 4) then take the complement of 10, i.e. 16, and this time L-leap down the square, filling in the complementar of 9, the number 17, the third arm of the cross, move up to 1 and to the last column to fill in the number 18, the fourth arm of the cross.
5. L-leap down and across the square to 20, jump to 21 and then back to 22 as shown. This completes the square.

the last three examples in the series (A, B and C) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares which places 5 into either of two corner cells. Note that in B the entry 5 goes to the empty left top cell not next to the 4:

Modified Method V using a modified Loubère square and a modified cross L-leap approach

A modified 3x3 De La Loubère square with a semi consecutive main diagonal along with the cross L-leap method is used to to fill in the boundary and generate an expanded magic square:

1. Construct a 3x3 Loubère non-magic square with the main diagonal reversed and incorporate into a larger 5x5 square.
2. Fill in the two corner cells to complete the main 5x5 diagonal.
3. Fill in the first cell in the fourth row with the number 4, since 3 was the last number on the complementary list and perform a cross L-leap, first across the square then up/down.
4. Using the required pair table below modify the cross L-leap and take the complement of 10, i.e. 16, but this time cross L-leap opposite to the number 8 and insert 16. cross L-leap across and fill in the first and last rows up to the number 19.
5. cross L-leap opposite to 4 insert a 20 then cross L-leap up the square filling in all the empty cells.

the last three examples in the series (D, E and F) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares and place the 5 into either of two corner cells. Note that in D, the number 5 goes to the bottom corner, while in E it goes to the top left corner:

Expanded 5x5 Loubère Squares using the Modified Method IV and V Approaches

In these examles 5x5 Loubère squares are expanded into 7x7 using the cross L-leap methods:

1. Construct a 5x5 Loubère square and incorporate into a larger 7x7 square. Fill in the two corner cells in normal fashion (non-reversed) to complete the main 7x78 diagonal. Fill in the sixth cell in the first row with the number 11, since 10 was the last number on the complementary list.
2. Perform a L-leap to the bottom right cell.
3. Carry out an L-leap up and across to 17, the first arm of the cross, then to 18 the second arm by way of 8 (the center of the cross) then back to 21, enter 29 its complement then perform an L-leap to 32 and the to 33 the third and fourth arms also by by way of 8.
4. Continue L-leaping down the square until all the empty cells are filled and fill in the last corner cell and finally 39.

three other examples in this series (G, H and I) are shown below after repeating the algorithm starting at the only three other positions on the square that produce magic squares which places 12 into either of two corner cells. Note that in G, the number 12 goes to the bottom corner, while in H and I it goes to the top left corner:

The last four examples in this series (J, K, L and M) shown below begin with the number 28 in the main diagonal and switch positions between 12 and 38:

this completes this section on new squares from modified wheel and De La Loubère methods. To continue the method of IA using 7x7 and 9x9 squares (Part IIA). To go back to Part IB of new Loubère squares. To return to homepage.