New Méziriac Full Pendulum Method (Part II)

A Discussion of the New Methods

An important general principle for generating odd magic squares by the Méziriac 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 Méziriac squares are shown below as examples:

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Méziriac 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 variable knight move as shown in the construction of the squares.

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

The new Méziriac squares, which I will label PMn^* (center cell#) [LD or RU,full arc] where PMn^* signifies a full arc pendulum move nxn Méziriac square with a certain center cell number, breaking either down or to the right. All squares in this group are magic except for those where n is divisible by three which are neither magic or semi-magic.

The new Méziriac Knight break squares, which I will label PMn^* (center cell#) [LD or RU,full arc,(n₁U,n₂L)] where PMn^* signifies a full arc pendulum move nxn Méziriac square with a certain center cell number, with a variable knight move breaking n₁ up then n₂ left. All squares in this group even where n is divisible by three are semi-magic.

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Construction of the 5x5 Méziriac Full Pendulum Magic Squares

5x5 Full Arc Squares

1. To generate the square, PM5^* 15 [RU, full arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancingcontinuously diagonally up right, then continuously down left until blocked by a previous number.
2. Move two cells down.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

1. To generate the magic square, PM5^* 14 [LD, full arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally down left, then continuously up right until blocked by a previous number.
2. Move two cells 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 semi-magic square, PM23^* 12 [RU, full arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching until blocked by a previous number.
2. Move two cells down.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

1. To generate the semi-magic square, PM7^* 24 [DL, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing continouosly diagonally down left, then continuously up right until blocked by a previous number.
2. Move two cells right.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

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

1. To generate the square, PMK5^* 15 [RU, full arc,(1U,2L)], place a 1 to the right of the center cell 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. For the knight break move one cell up two cells left.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

1. To generate the square, PMK5^* 15 [RU, full arc,(2U,1L)], place a 1 to the right of the center cell 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. For the knight break move two cells up one cell left.
3. Repeat the process until the square is filled, as shown below in squares 1-5.

Construction of 7x7 Knight-Break Méziriac Full Arc Pendulum Magic Squares

1. To generate the square, PMK7^* 25 [RU, full arc,(2U,3L)], place a 1 to the right of the center cell 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. For the knight break move two cells up three cells left.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

1. To generate the square, PMK7^* 28 [RU, full arc,(3U,2L)], place a 1 to the right of the center cell 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. For the knight break move three cells up two cells left.
3. Repeat the process until the square is filled, as shown below in squares 1-4.

This completes this section on Méziriac full pendulum squares (Part II). To return to homepage.