A New Set of Bachet de Méziriac Methods and Squares (Part IV)
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 and
Méziriac method Part II, and Méziriac method Part III.
As stated above two of the pairs are complementary and two are noncomplementary. therefore, two squares have center cells that are
½(n^{2} + 1) and each of the other two are either
½(n^{2} + n  2) or
½(n^{2}  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 semimagic squares as in
nonregular Bachet de Méziriac squares.
Construction of the Regular and Nonregular 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 semimagic squares.
Break and 3 cells to the right
I Magic
25  7  19 
1  13 
6  18  5 
12  24 
17  4  11 
23  10 
3  15  22 
9  16 
14  21  8 
20  2 


II Magic
21  8  20 
2  14 
7  19  1 
13  25 
18  5  12 
24  6 
4  11  23 
10  17 
15  22  9 
16  3 


III Magic
22  9  16 
3  15 
8  20  2 
14  21 
19  1  13 
25  7 
5  12  24 
6  18 
11  23  10 
17  4 


IV Magic
23  10  17 
4  11 
9  16  3 
15  22 
20  2  14 
21  8 
1  13  25 
7  19 
12  24  6 
18  5 


V Magic
24  6  18 
5  12 
10  17  4 
11  23 
16  3  15 
22  9 
2  14  21 
8  20 
13  25  7 
19  1 

Break and 3 cells down
I SemiMagic
22  10  18 
1  14 
9  17  5 
13  21 
16  4  12 
25  8 
3  11  24 
7  20 
15  23  6 
19  2 


II Magic
23  6  19 
2  15 
10  18  1 
14  22 
17  5  13 
21  9 
4  12  25 
8  16 
11  24  7 
20  3 


III SemiMagic
24  7  20 
3  11 
6  19  2 
15  23 
18  1  14 
22  10 
5  13  21 
9  17 
12  25  8 
16  4 


IV SemiMagic
25  8  16 
4  12 
7  20  3 
11  24 
19  2  15 
23  6 
1  14  22 
10  18 
13  21  9 
17  5 


V SemiMagic
21  9  17 
5  13 
8  16  4 
12  25 
20  3  11 
24  7 
2  15  23 
6  19 
14  22  10 
18  1 

From this table it is seen that all four squares in one group of (IV) 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
Equation  Value Right  S_{r}
 Equation  Value Down  S_{left}+d2  d2  Value d2 
½(n^{2}  3)  11  65  ½(n^{2}  1)  12  60  n  5 
½(n^{2}  1)  12  65  ½(n^{2} + 1)  13  65  0  0 
½(n^{2} + 1)  13  65  ½(n^{2} + 3)  14  70  n  5 
½(n^{2} + 3)  14  65  ½(n^{2} + 5)  15  75  2n  10 
½(n^{2} + 5)  15  65  ½(n^{2}  3)  11  55  2n  10 
Results of 25x25 Méziriac type Squares
the next squares exhibiting these pentad properties are those 25x25 magic and semimagic 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 S_{left} 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  S_{left}+d2  Value d2 
301  306  311  316  321  7800  n 
302  307  312  317  322  7825  0 
303  308  313  318  323  7850  n 
304  309  314  319  324  7875  2n 
305  310  315  320  325  7775  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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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 
48  32  16 
7  40 
24  8 
48  32  16 
7  40 
24  8 
31  15  6 
39  23 
14  47 
31  15  6 
39  23 
14  47 
21  5  38 
22  13 
46  30 
21  5  38 
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
MéziriacKnight type Squares.
To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com