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

A Discussion of the New Methods

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

1. Every number on the main diagonal is represented at least once in this type of square.
2. For Loubère Knight-step where n is divisible by 3, i.e., 3(2n + 1) n squares are produced where the sum of the left diagonal S equals S + d, and d is equal to either -n², 0 or n².

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

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 fashion down until blocked by a previous number. One square shows a typical knight and break move.
2. Move one cell up.
3. Repeat the process until the square is filled.
4. Generate the five squares belonging to a group.

The 5x5 Loubère Knight-step Squares

The Set of 5x5 Knight-step Loubère Magic Square [(2D,1L),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 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 a group.

The 5x5 Loubère Knight-step Squares

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 showing the triad sums of the left diagonal. Squares K and L show a typical knight and up or down moves.

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 LK7^* 39 [(2D,1L),1R] one can move up the right diagonal on a plane of four LK7^* 39 [(2D,1L),1R] 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 II). The next section deals with a new Loubère Knight-step method (Part III). To return to homepage.