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

A Discussion of the New Methods

An important general principle for generating odd magic squares by the De La Loubère method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). The properties of these regular or associated Loubère squares are:

1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n² + 1, i.e., or twice the number in the center cell and are complementary to each other.

The 5x5 and 7x7 regular Loubère squares are shown below as examples:

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Loubère squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time. In this new method each subsequent number is added in a stepwise knight step manner, followed by a right or an up break. In addition, n odd squares may be constructed with the initial number 1 on either of two broken diagonals shown below in light blue or yellow separated by symmetry by a light grey diagonal for a 5x5 square.

The new Loubère type squares of Part I, which I will label KLn^* (center cell#) [(2D,1L), U or R] where KLn^* signifies a Knight-step nxn Loubère square with the center cell number of the square and breaking either to the right or up.

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.

Construction a 5x5 Knight-step Loubère Magic Square

5x5 Squares

1. To generate the regular square, KL5^* 8 [(2D,1L),1R], place a 1 into the center of the first row of a 5x5 square and fill in cells by advancing in a knight fashion to the right until blocked by a previous number.
2. Move one cell right.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

The Other Four 5x5 Loubère Knight-step Squares

A 7x7 Loubère Knight-step Square

The construction of a 7x7 Loubère Knight-step square KL7^* 39 [(2D,1L),1R] from the broken yellow diagonal is shown below using the knight-step approach.

Three 9x9 Loubère Knight-step Squares

Three 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. Square A shows a typical knight and Right move.

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^* 39 [(2D,1L),1R] one can move up the right diagonal on a plane of four KL7^* 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 two step squares (Part I). The next section deals with a new Loubère Knight-step square method (Part II). To return to homepage.