A New True Bachet de Méziriac Variable Knight Method and Squares (Part II)

Regular and Non-Regular Méziriac Variable Knight Combo Magic and Semi-Magic Squares

A Discussion of the New Methods

An important general principle for generating odd magic squares by the Bachet de 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 Bachet de Méziriac 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.
3. The same regular square is produced when the initial 1 is placed on the center of the first row or the center of the last column, however, this is not the case for the first column or last row.

However, for the Bachet de Méziriac methods two regular squares are produced for 5x5 squares, while one regular square is produced for a Loubère square as shown below:

In continuation from the previous page Part I we have shown one form of a knight move. The second knight move is similar to the first except that after a break one moves in a knight fashion ½(n - 1) moves down followed by ½(n - 3) moves right for the broken yellow diagonal . Similarly, for the broken light blue diagonal we perform ½(n - 1) moves left followed by ½(n - 3) moves up. This produces one regular square and n - 1 non-regular squares. Squares belonging to one diagonal group are identical to one in the other diagonal group. So that for example a square starting with the number 1 in the first column is identical to that with the number 1 in the last row. Using the 7x7 square as an example shows the two diagonals and typical 1 positions. The second table shows the equation and value of the center cell of each square for the light blue diagonal. Starting with the square generated from 1 in the first row the values range from ½(n² - n + 2) to ½(n² + 7) for squares breaking to the left.

These new Bachet de Méziriac type squares where hte L1 and L2 lengths of the knight move are interchanged and which I will label MZK7^* (center cell#) (K[L1,L2] or K[L3,L4]) where (MZK7^* signifies a true 7x7 Méziriac-Knight square with the center cell number of the square and a break followed by:

1. a knight move L1= (½(n - 1) cell Down,L2= ½(n - 3) cells Right) for the yellow broken diagonal squares or
2. a knight move L3= (½(n - 1) cell Left,L4= ½(n - 3) cells Up) for the light blue broken diagonal.

The squares exhibit the following properties:

1. Every number on the main diagonal is represented at least once in this type of square.
2. No odd squares divisible by 5, i.e., 3(2n + 1) are possible by this method, although these squares do have some partial magic properties.
3. Odd squares divisible by 3 but not 5 are magic or semi-magic.

Construction of Regular and Non-Regular Méziriac-Knight Squares

Examples of 7x7 Squares

For the first square MZK7^* 24 K[3L,2U]:

1. In a 7x7 square place the number 1 at the end of the second row 7x7 and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
2. Move three cell to the left and two cells up.
3. Repeat the process until the square is filled, as shown below in squares 1-3.

Note that this squares is non-regular. The following complete three 7x7 squares from the 1 on the broken light blue diagonal show 1 regular and 2 non-regular squares.

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The complementary tables for the last three 7x7 squares may be seen on new bachet squares (Part II) and new bachet squares (Part III).
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Examples of 9x9 Squares

The following are examples of regular and non-regular 9x9 square from the broken blue diagonal showing the 8^th knight move:

The 9x9 and 21x21 Value tables

The following include the 9x9 and 21x21 center cell values and the magic sum S of the left main diagonal. These odd n divisible by 3 but not 5 cycle through the triad of d2 values -n, 0 and n and the yellow entries are magic and the others semi-magic:

The Plane of Loubère-Knight 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 MZK7^* 25 K[3L,2U] one can move up the right diagonal on a plane and generate the complete set of 7 squares as is shown in Part IV of the Bachet de Méziriac series.

This completes this section on the new Bachet de Méziriac Knight method and squares (Part II). To go back to Part I or to return to homepage.