A New Set of Bachet de Méziriac Methods and Squares (Part II)
Regular and NonRegular 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(n^{2} + 1). The properties of these regular or associated Bachet de Méziriac squares are:
 That the sum of the horizontal rows,
vertical columns and corner diagonals are equal to the magic sum S.
 The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to
n^{2} + 1, i.e., or twice the number in the center cell and are complementary to each other.
 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. Note that
MT7^{*} (center cell#) (RIGHT or LEFT) signifies a 7x7 Méziriac type square with the center cell number of
the square and breaking either to the right or to the left. Construction of all the squares on the nonmain diagonals
will continue on the next two pages: Méziriac method Part III and Méziriac method Part IV.
Construction of Méziriac type Squares
Second Set of Four 7x7 Squares
For the first MT7^{*} 24 RIGHT square:
 Place the number 1 in cell 3 of row 3 of a 7x7 square and fill in cells by advancing diagonally upwards to the right
until blocked by a previous number.
 Move three cells right.
 Repeat the process until the square is filled, as shown below in squares 13.
1nr
11   
19  3 
 
  18 
2  
 10 
 17  1 
 
9  
16  7  
 8 
 
6   
14  
 15 
  13 
 
 5 
 12  
 
4  

⇒ 
2nr
11   
19  3 
 27 
  18 
2  
26  10 
 17  1 
 25 
9  
16  7  
24  8 
 
6   23 
14  
 15 
 22  13 
 
21  5 
28  12  
29  20 
4  

⇒ 
3nr
11  44  35 
19  3 
36  27 
43  34  18 
2  42 
26  10 
33  17  1 
41  25 
9  49 
16  7  40 
24  8 
48  32 
6  39  23 
14  47 
31  15 
38  22  13 
46  30 
21  5 
28  12  45 
29  20 
4  37 

For the second MT7^{*} 26 DOWN square:
 Place the number 1 in cell 3 of row 3 of a 7x7 square and fill in cells by advancing diagonally upwards to the right
until blocked by a previous number.
 Move three cells down.
 Repeat the process until the square is filled, as shown below in squares 13.
 To keep the colored square simple and uncluttered only the three pairs of colors in the beginning of each group are shown.
1nr
14   
18  3 
 
  17 
2  
 13 
 16  1 
 
12  
15  7  
 11 
 
6   
10  
 
  9 
 
 5 
 8  
 19 
4  

⇒ 
1nr
14   
18  3 
 22 
  17 
2  
28  13 
 16  1 
 27 
12  
15  7  
26  11 
 
6   25 
10  
29  21 
 24  9 
 
20  5 
23  8  
 19 
4  

⇒ 
2nr
14  48  33 
18  3 
37  22 
47  32  17 
2  36 
28  13 
31  16  1 
42  27 
12  46 
15  7  41 
26  11 
45  30 
6  40  25 
10  44 
29  21 
39  24  9 
43  35 
20  5 
23  8  49 
34  19 
4  38 

Note that fourteen pairs sum to 47 and ten pairs sum to 54 including the center square when multiplied by 2.
In addition, none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table.
Also this complementary table is identical to the 7x7 semisquare in nonregular Loubère squares.
For the third MT7^{*} 25 RIGHT square:
 Place the number 1 in cell 2 of row 4 of a 7x7 square and fill in cells by advancing diagonally upwards to the right
until blocked by a previous number.
 Move three cells right.
 Repeat the process until the square is filled, as shown below in squares 13.
1
12   
 4 
 
  19 
3  
 11 
 18  2 
 
10  
17  1  
 9 
 
7   
8  
 16 
  14 
 
15  6 
 13  
 
5  

⇒ 
2
12   29 
20  4 
 28 
  19 
3  
27  11 
 18  2 
 26 
10  
17  1  
25  9 
 
7   24 
8  
 16 
 23  14 
 
15  6 
22  13  
 21 
5  

⇒ 
3
12  45  29 
20  4 
37  28 
44  35  19 
3  36 
27  11 
34  18  2 
42  26 
10  43 
17  1  41 
25  9 
49  33 
7  40  24 
8  48 
32  16 
39  23  14 
47  31 
15  6 
22  13  46 
30  21 
5  38 

Note that three pairs sum to 56 including the center square when multiplied by 2 and twentyone pairs sum to 49.
In addition none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table. Also
his complementary table is identical to the one in nonregular Loubère squares
For the fourth MT7^{*} 27 DOWN square:
 Place the number 1 in cell 2 of row 4 of a 7x7 square and fill in cells by advancing diagonally upwards to the right
until blocked by a previous number.
 Move three cells down.
 Repeat the process until the square is filled, as shown below in squares 13.
 To avoid cluttering only the first color in the group is shown.
1
8   
19  4 
 
  18 
3  
 14 
 17  2 
 
13  
16  1  
 12 
 
7   
11  
 15 
  10 
 
 6 
 9  
 
5  

⇒ 
2
8   
19  4 
 23 
  18 
3  
22  14 
 17  2 
 28 
13  
16  1  
27  12 
 
7   26 
11  
 15 
 25  10 
 29 
21  6 
24  9  
 20 
5  

⇒ 
3
8  49  34 
19  4 
38  23 
48  33  18 
3  37 
22  14 
32  17  2 
36  28 
13  47 
16  1  42 
27  12 
46  31 
7  41  26 
11  45 
30  15 
40  25  10 
44  29 
21  6 
24  9  43 
35  20 
5  39 

Note that seven pairs sum to 45 and seventeen pairs sum to 52 including the center square when multiplied by 2.
In addition, none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table.
Also this complementary table is identical to the one in nonregular Loubère squares.
This completes this section on new Bachet de Méziriac method and squares. To proceed to Part III or to return to
homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com