A New Set of Bachet de Méziriac Methods and Squares (Part IV)

Regular and Non-Regular Bachet de Méziriac Squares

A Break Followed by three Moves

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.

this web page is a continuation of the previous Méziriac method Part I and Méziriac method Part II, and Méziriac method Part III.

As stated above two of the pairs are complementary and two are non-complementary. therefore, two squares have center cells that are ½(n² + 1) and each of the other two are either ½(n² + n - 2) or ½(n² - n + 4), respectively. In addition, and very importantly, all the odd squares are magic except for odd squares divisible by 3, i.e., 3(2n + 1), which are not magic. If we use the 5x5 squares as an example, we find that the squares produced by this method are identical to those produced as in two of the examples above If we place the number 1 in row 2 above center or 4 below center. Placing the 1 to the right or left of center produces semi-magic squares as in non-regular Bachet de Méziriac squares. ********************************************************************************************************************************************************

Construction of the Regular and Non-regular Méziriac type Squares

the 5x5 Squares

Although we started above with 7x7 squares which are all magic, below are shown examples of those odd squares divisible by 5 but not by 3 such as the 5x5. the center Value table shows the value of each center cell and the last two columns gives the value of delta obtained from S + d2, the sum of the left diagonal. Because 3 moves to the right or down is the same as 3 moves left or up for the 5x5 square, it is seen that these squares are the only ones that show this property. the point I am trying to make is that larger squares divisible by 5 and not by 3, of which the 25x25 is the next example after the 5x5, also behave similarly, i.e., consist of magic and semi-magic squares.

From this table it is seen that all four squares in one group of (I-V) are magic while only one in the other group is magic. In addition d2 ranges from -2n to 2n, i.e., the pentad -2n, -n, 0, n, 2n, which *************************************************************************************************************************************************************

this completes this section on the Bachet de Méziriac method and squares. the next page gives a new method for the construction of Méziriac-Knight type Squares. To return to homepage.