New Squares from the Modified De La Loubère Method

Part IIA

Further Discussion of New De La Loubère type squares

The previous page showed how to use Modified De La Loubère squares of order 3 to prepare larger L-leap and wheel add ons. This page will show how to generate larger Loubère squares where the internal square is 5x5 expanded to 7x7 or a 7x7 expanded into a 9x9. The first set of squares will show one of the exceptions to the rule

1. Incorporate the 5x5 grey square (starting with the number 7) using the modified Loubère method into a larger 7x7 square as shown below.
2. Fill in the two corner cells to complete the main diagonal.
3. Begin a L-leap on the boundary from the unused numbers in the 7x7 complementary table.
4. Stop at the first cell in the second column inserting a 5, leap across to the right lower corner and insert a 6, then leap to the second cell in the first column and insert a 17, since all numbers from 7 to 16 have been used to generate the 5x5 square.
5. Continue L-leaping right and left across until the first cell in the sixth row is reached.
6. Take the complement of 21, i.e. 29, and this time L-leap backwards across the square, filling in all the complementary cells up to the left corner cell in the first column.
7. L-leap down and up filling in all the cells with the right complementary numbers until the square is completed.
8. At this point columns 1 and 7 do not total to the magic sum 175. Interchange these two numbers and the square is now magic.

Square group 9, i.e., beginning with the number 9 also behaves similarly. With the 30 and 20 in this position it is possible to move pairs of adjacent numbers around the boundary and still retain the magic. Remember that there are more than one magic squares in each group since any adjacent pair may be moved about on the boundary.

The next series of 7x7 squares expanded into 9x9s show two adjacent groups 10 and 11 where the former turns out to be magic and the latter not since the first and last columns add up to 390 and 348, respectively instead of 369. The only difference between the boundary numbers is that 72 and 10 replace 51 and 31, respectively. In addition, the way the boundary is set up an odd number follows an even number on the same line or column (except on the corners in some cases) for the square to be magic. As can be seen this rule is broken since 32 follows 72 and 50 follows 10.

A 5x5 Loubère Square Expanded into a 7x7 using the >Wheel Boundary Method

1. Incorporate a # 2 group 5x5 Loubère modified square into a 7x7 square, where the blue colors correspond to a shift from low to high complementary numbers.
2. Fill in the two corner cells to complete the main diagonal.
3. Fill in the six boundary cells with a subset of three pairs of spoke numbers.
4. Fill in the top and bottom non-spoke cells with the requisite pair of numbers in reverse order so that the columns and rows total to 175.
5. Fill in the right and left non-spoke cells with the requisite pair of numbers in reverse order so that the columns and rows total to 175.

Attempt to Expand a 5x5 Loubère Square into a 7x7 using the Wheel Boundary Method

1. Incorporate a 5x5 Loubère modified square into a 7x7 square, where the blue colors correspond to a shift from low to high complementary numbers.
2. Fill in the two corner cells to complete the main diagonal.
3. Cannot perform a spoke wheel boundary addition due to the lone pair (1,19) which must be paired off e.g. (1,49) and (2,48) adjacent pairs (see discussion of wheel method rules . This group of squares along with group square 11 (starting position 11) produce no expanded magic squares.

This completes this section on new squares from the expanded De La Loubère methods. To continue with Part IB using 7x7 squares (Part IIB). To see previous method Part IC: Loubère Square using 3x3 and 5x5 squares or to return to homepage.