New Loubère Full Pendulum Method (Part I)

A Discussion of the New Method

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 using the full pendulum approach. When a break is encountered this may be a single move (right or down) or a knight move as shown in the construction of the squares.

1. Fill in the starting number 1 and add the next numbers consecutively down the ladder followed by consecutively up the ladder until the broken diagonal is filled tracing out a full arc of a pendulum or
2. Fill in the starting number 1 and add the next numbers consecutively up the ladder followed by consecutively down the ladder until the broken diagonal is filled again tracing out a full arc of a pendulum.

In the first group of new Loubère squares, which I will label PLn^* (center cell#) [LD or RU,full arc] where PLn^* signifies a full arc pendulum move nxn, left down or up right, Loubère square with the center cell number of the square and breaking either down or to the right. In addition, all odd squares, except those divisible by three, are magic.

In the second group the squares are labeled PLKn^* (center cell#) [LD or RU,full arc,(n₁,n₂)] where PLKn^* signifies a full arc pendulum move nxn Loubère square with a center cell number and breaking in knight fashion. (n₁, n₂) may be either (1 up, 2 right) or (2 down, 1 left). In addition, all odd squares, except those divisible by three, are magic.

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Construction of the 5x5 Loubère Full Pendulum Magic Squares

5x5 Full Arc Squares

1. To generate the square, PL5^* 14 [LD, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively first down left filling the partial diagonal, then up right consecutively filling the rest of the diagonal until blocked by a previous number.
2. Move one cell down.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

1. To generate the magic square, PL5^* 13 [RU, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively first up then right, filling up the broken diagonal 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.

7x7 Full Arc Squares

1. To generate the magic square, PL7^* 24 [LD, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively down left, then consecutively down left until blocked by a previous number.
2. Move one cell down.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

1. To generate the magic square, PL7^* 26 [RU, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively up right, then consecutively down left 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 1-4.

Construction of 5x5 Knight-Break Loubère Full Arc Pendulum Magic Squares

These new Loubère squares, which I will label PLKn^* (center cell#) [LD or RU,small or large arc,(U,R or D,L)] where PLKn^* signifies a small or large arc pendulum move. nxn is a Loubère square with a certain cell number, breaking in knight fashion (up,right) or (down,left)

1. To generate the magic square, PLK5^* 15 [LD,full arc,(1U,2R)], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively down left, then consecutively up right until blocked by a previous number.
2. Move in knight fashion one cell up and two cells right.
3. Repeat the process until the square is filled, as shown below in squares 1-5.
4. Note that PLK5^* 15 [LD,full arc,(2D,1L)] does not generate a magic square.

1. To generate the magic square, PLK5^* 12 [RU,full arc,(2D,1L)], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively up right, then consecutively left down until blocked by a previous number.
2. Move in knight fashion two cells down and one cell left.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

1. To generate the magic square, PLK7^* 23 [LD,full arc,(1U,2R)], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally consecutively down left, then consecutively up right until blocked by a previous number.
2. Move in knight fashion one cell up and two cells right.
3. Repeat the process until the square is filled, as shown below in squares 1-5.
4. Note that PLK7^* 26 [LD,full arc,(2D,1L)] in this case generates a magic square.

1. To generate the magic square, PLK7^* 27 [RU,full arc,(2D,1L)], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left until blocked by a previous number.
2. Move in knight fashion two cells down and one cell left.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

This completes this section on De La Loubère full pendulum squares (Part I). The next section deals with new Méziriac full pendulum squares method (Part II). To return to homepage.