New De La Loubère Knight-step Method and Squares (Part III)

A Discussion of the New Methods

This is a continuation of Part II which details the set of Loubère Knight-step squares. The new Loubère type squares of Part III, KLn^* (center cell#) [knight move,1U or 1D] where KLn^* signifies a Knight-step nxn Loubère type square with the center cell number of the square, with a knight move of (2U,1R) or (2D,1R), followed by breaking either up or down. Note that these apply to those squares generated with 1 on the yellow diagonal see Part I.

This method differs from the previous due to the two ways the knight may move. It may move down 2, right 1, i.e., (2D,1R) or up 2, right 1, i.e., (2U,1R).

1. Every number on the main diagonal is represented at least once in this type of square.
2. For every square not divisible by 3, the (2U,1R) set forms magic squares for the left diagonals, while the (2D,1R) forms semi-magic squares in the left diagonal.
3. For Loubère Knight-step where n is divisible by 3, i.e., 3(2n + 1), the sum of the left diagonal is always magic for the (2D,1R) squares and semi-magic for the (2U,1R) squares.

The Set of 5x5 Knight-step Loubère Magic Squares [(2U,1R),1D]

1. To generate these square, place a 1 into any of the cells on the yellow diagonal of a 5x5 square and fill in cells by advancing in a knight (2 cells up,1 cell right) fashion down until blocked by a previous number. One square shows a typical knight and break move.
2. Move one cell down.
3. Repeat the process until the square is filled.
4. Generate the five squares belonging to the group.

The 5x5 Loubère Knight-step Magic Squares

The Set of 5x5 Knight-step Loubère Semi-magic Squares [(2D,1R),1D]

1. To generate these square, place a 1 into any of the cells on the yellow diagonal row of a 5x5 square and fill in cells by advancing in a knight (2 cells down,1 cell right) fashion down until blocked by a previous number. One square shows a typical knight and break move.
2. Move one cell down.
3. Repeat the process until the square is filled.
4. Generate the five squares belonging to the group having left diagonal sums of 65, 55, 70, 60 and 75, respectively, where the last four sums are semi-magic.

The 5x5 Loubère Knight-step Semi-magic Squares

Two 7x7 Loubère Knight-step Squares

Two 7x7 Loubère Knight-step square from the broken yellow diagonal are shown. Square K belongs to the set of (2U,1R) and L belongs to (2D,1R) set. Typical knight and break moves are shown.

Two 9x9 Loubère Knight-step Squares

Two 9x9 Loubère Knight-step square from the broken yellow diagonal are shown along with the accompanying table of 9 squares. The first square is composed of triad sums for the left diagonal, while the second square is magic for each of the left diagonals. Squares K and L show a typical knight and up or down moves. Square M belongs to the set of (2U,1R) and N belongs to (2D,1R) set. Typical knight and break moves are also shown.

The Plane of Loubère Squares

At this point it may be said that alternatively these squares may be constructed using a plane of four squares. For example using the 7x7 square KL7^* 25 [(2U,1D),1D] one can move up the right diagonal on a plane of four KL7^* 39 [(2U,1D),1D] and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

This completes this section on regular and non-regular De La Loubère knight-step squares (Part III). To visit the new Knight version of Méziriac squares (Part I) or To return to homepage.