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 Méziriac square

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.

Break and 3 cells to the right
I Magic
25 7 19 1 13
6185 12 24
17411 23 10
31522 9 16
14 21 8 20 2
 
II Magic
21 8 20 2 14
7191 13 25
18512 24 6
41123 10 17
15 22 9 16 3
 
III Magic
22 9 16 3 15
8202 14 21
19113 25 7
51224 6 18
11 23 10 17 4
 
IV Magic
23 10 17 4 11
9163 15 22
20214 21 8
11325 7 19
12 24 6 18 5
 
V Magic
24 6 18 5 12
10174 11 23
16315 22 9
21421 8 20
13 25 7 19 1
Break and 3 cells down
I Semi-Magic
22 10 18 1 14
9175 13 21
16412 25 8
31124 7 20
15 23 6 19 2
 
II Magic
23 6 19 2 15
10181 14 22
17513 21 9
41225 8 16
11 24 7 20 3
 
III Semi-Magic
24 7 20 3 11
6192 15 23
18114 22 10
51321 9 17
12 25 8 16 4
 
IV Semi-Magic
25 8 16 4 12
7203 11 24
19215 23 6
11422 10 18
13 21 9 17 5
 
V Semi-Magic
21 9 17 5 13
8164 12 25
20311 24 7
21523 6 19
14 22 10 18 1

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, with

Center Cell Value
EquationValue Right    Sr     EquationValue DownSleft+d2d2Value d2
½(n2 - 3)1165½(n2 - 1)1260-n-5
½(n2 - 1)1265½(n2 + 1)136500
½(n2 + 1)1365½(n2 + 3)1470n5
½(n2 + 3)1465½(n2 + 5)15752n10
½(n2 + 5)1565½(n2 - 3)1155-2n-10

Results of 25x25 Méziriac type Squares

the next squares exhibiting these pentad properties are those 25x25 magic and semi-magic squares that break in the down direction (those that break right are all magic). If we tabulate the center numbers of each of the 25x25 squares we get the table, where Sleft is the sum of the main left diagonal and the yellow entry signifies that all the squares having those centers on that line are magic:

25x25 Square Values
center of Square Sleft+d2Value d2
3013063113163217800-n
30230731231732278250
3033083133183237850n
30430931431932478752n
3053103153203257775-2n

The Plane of Méziriac type Squares

It is a good time to say at this point that an alternative method for the construction of every square in a set of Méziriac type squares (those that break to the right or to the left) is possible. For example, we can construct a finite plane of a repeated square, e.g., MT7* 22 RIGHT and use this plane of numbers to generate all the other squares in ther set. (Note that MT7* signifies a 7x7 Méziriac type with a break followed by moving 3 cells to the right). the next squares in the set MT7* 23 RIGHT and MT7* 24 RIGHT are then generated by moving up the main right diagonal as shown below:

A Plane of four MT7* 22 RIGHT SQUARES
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42
9 49 33 17 1 41 25 9 49 33 17 1 41 25
483216 7 40 24 8 483216 7 40 24 8
31156 39 23 14 47 31156 39 23 14 47
21538 22 13 46 30 21538 22 13 46 30
4 37 28 12 45 29 20 4 37 28 12 45 29 20
36 27 11 44 35 19 3 36 27 11 44 35 19 3
26 10 43 34 18 2 42 26 10 43 34 18 2 42

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.


Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com