New Bachet de Méziriac Method and Semi-Magic Squares (Part II)

Regular and Non-Regular Bachet de Méziriac Semi-Magic Squares

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 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.
  3. the same regular square should be produced when the initial 1 is placed to the top, right, left or bottom of the center cell.

However, for the Bachet de Méziriac methods the four regular squares produce two equal pairs as shown below for 5x5 and 7x7 squares:

3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23
  
22 9 16 3 15
8202 14 21
19113 25 7
51224 6 18
11 23 10 17 4
 
4 29 12 37 20 45 28
351136 19 44 27 3
104218 43 26 2 34
411749 25 1 33 9
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46
 
45 20 37 12 29 4 28
193611 45 3 27 44
421034 2 26 43 18
9331 25 49 17 41
32 7 24 48 16 40 8
6 23 47 15 39 14 31
22 46 21 38 13 30 5

this web page has been modified from the previous to take into account new findings since the previous Bachet de Méziriac method has been found to be a subset of a general method. It will be shown that the initial numeral 1 can be placed anywhere on two broken diagonals, in which the original Méziriac method is placed on the bottom diagonal and the the second initial 1 is placed on the top diagonal. this page will show that all the squares having an initial 1 on the broken diagonal are related to one another. It will also be shown that both regular and non-regular squares are produced. the squares on the left are produced by a method which give rise to semi-magic squares unlike the previous method which gave rise to magic squares.

All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the light grey main diagonal behave differently from typical Loubère squares. After a break/2 moves right for the broken yellow diagonal and break/2 moves up for the broken light blue diagonal, one regular square and n - 1 non-regular squares are produced. Squares belonging to one diagonal group are identical to another square on the other diagonal group. 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 (starting with the square generated from 1 in the first row) where the values range from ½(n2 - n + 2) to ½(n2 + 7) for squares breaking right or up.

The set of Broken Diagonals
1 1
1 1
1 1
1 1
1 1
1 1
1 1
  
center Value
EquationValue LeftEquationValue Down
½(n2 - 3)23½(n2 - 5)22
½(n2 - 1)24½(n2 - 3)23
½(n2 + 1)25½(n2 - 1)24
½(n2 + 3)26½(n2 + 1)25
½(n2 + 5)27½(n2 + 3)26
½(n2 + 7)28½(n2 + 5)27
½(n2 - 5)22½(n2 + 7)28

These new Méziriac squares, which I will label Mn* (center cell#) (RIGHT or UP) where (Mn* signifies a nxn Méziriac square with the center cell number of the square and breaking either right or up. this would make the original Méziriac squares depicted in the introduction as M5* 13 RIGHT and M7* 13 RIGHT. the squares exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. Of the odd squares divisible by 3, i.e., 3(2n + 1) only one is magic.
  3. Of the odd squares divisible by 5 but not 3 only one of which is magic, the rest semi-magic.

Thus, these new squares are either regular or non-regular Bachet de Méziriac semi-magic squares.

Construction of Regular and Non-regular Bachet de Méziriac Semi-magic Squares

An example of a 5x5 Square

  1. Place a 1 below the center cell of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move two cells right, when a block is encountered. (the alternative method is to place a 1 to the left of center and move to the up 2 cells when a block is encountered.)
  3. Repeat the process until the square is filled, as shown below in squares 1-2.
1
4 10 11
9 3
8 2
1 7
5 6
2
4 17 10 23 11
16922 15 3
82114 2 20
25131 19 7
12 518 6 24
A five series

Note that the five pairs sum to 23 and seven pairs sum to 28 including the center square when multiplied by 2 and the sum of the left diagonal is 65 + 5= 70. In addition none of these pairs are complementary, as in the regular Bachet de Méziriac squares, as shown in the connectivities of the complementary table.

The Four Other 5x5 semi-magic Squares

the following four squares complete the set of 5x5 from the yellow broken diagonal :

I
2 20 8 21 14
19725 13 1
62412 5 18
23114 17 10
15 3 16 9 22
 
II Magic
3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23
 
III
5 18 6 24 12
171023 11 4
92215 3 16
21142 20 8
13 1 19 7 25
 
IV
1 19 7 25 13
18624 12 5
102311 4 17
22153 16 9
14 2 20 8 21

7x7 Square Examples

The following 7x7 regular Bachet de Méziriac square was constructed using a 1 from the light blue diagonal.

 
1
10 4
15 9 3
8 2
14 1
7 13
6 12
11 5
2
16 10 4 22
15 9 3
8 2 21
14 1 20
7 19 13
6 18 12
17 11 5
3
47 16 41 10 35 4 22
15409 34 3 28 46
39833 2 27 45 21
14321 26 44 20 38
31 7 25 43 19 37 13
6 24 49 18 36 12 30
23 48 17 42 11 29 5

Note that the seven pairs sum to 45 and seventeen pairs sum to 52 including the center square when multiplied by 2 and the sum of the left diagonal is 175 + 7 = 182. In addition none of these pairs are complementary, as in the regular Bachet de Méziriac squares, as shown in the connectivities of the complementary table.

A seven series

The following four 7x7 squares are a sampling of the set from the light blue broken diagonal:

I M7* 24 UP
45 21 39 8 33 2 27
203814 32 1 26 44
371331 7 25 43 19
12306 24 49 18 36
29 5 23 48 17 42 11
4 22 47 16 41 10 35
28 46 15 40 9 34 3
 
II M7* 27 UP
48 17 42 11 29 5 23
164110 35 4 22 47
40934 3 28 46 15
8332 27 45 21 39
32 1 26 44 20 38 14
7 25 43 19 37 13 31
24 49 18 36 12 30 6
III M7* 22 UP
43 19 37 13 31 7 25
183612 30 6 24 49
421129 5 23 48 17
10354 22 47 16 41
34 3 28 46 15 40 9
2 27 45 21 39 8 33
26 44 20 38 14 32 1

The Plane of Méziriac 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 M7* 22 UP one can move up the right diagonal on a plane of four M7* 22 UP and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

this completes this section on non-regular Bachet de Méziriac semi-magic squares. To return to homepage.


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