A New True Bachet de Méziriac Variable Knight Method and Squares (Part I)
Regular and NonRegular Méziriac Variable Knight Combo Magic Squares
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.
However, for the Bachet de Méziriac methods two regular
squares are produced for 5x5 squares, while one regular square is produced for a Loubère square as shown below:
3  16  9 
22  15 
20  8  21 
14  2 
7  25  13 
1  19 
24  12  5 
18  6 
11  4  17 
10  23 


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 


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

A new set based on the true, original Bachet de Méziriac square is now possible. The difference with this knight method and previous ones is that
the knight moves take on a dynamic or variable character, meaning that as
the order of the square n increases as the length of the knight moves increases. In addition two possible knight moves are possible
both going in the same direction. The next page (Part II) will show the second type of knight move.
Previous Loubère and Méziriac type
knight methods employed a regular knight move (2 moves followed by a 1 move). Because the left main diagonal and the broken diagonals of a true Méziriac square are
composed of arithmetic progressions only the 5x5 squares can be constructed using the knight equation from above, i.e., (2,1) moves. Construction of higher order squares,
it appears, require a dynamic knight square move to produce magic squares. Although the new broken diagonals generated are not geometrical
they are very close to it.
So to continue all odd squares having the numerical 1's lying on two broken diagonals symmetrical with the
light grey main diagonal behave differently from typical
Méziriac squares. After a break one moves in a knight fashion ½(n  3) moves down followed by
½(n  1) moves right for the broken yellow diagonal . Similarly,
for the broken light blue diagonal we perform
½(n  3) moves left followed by ½(n  1) moves up.
This produces one regular square and n  1 nonregular squares. Squares belonging to
one
diagonal group are identical to one in the other diagonal group. So that for example a square starting with the number 1
in the first column is identical to that with the number 1 in the last row.
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
for the light blue diagonal. Starting with the square generated from 1 in the first row the values range from
½(n^{2}  n + 2) to ½(n^{2} + 7) for squares breaking
to the left.
The set of (Break + knight) Diagonals
1  

 
1  
  
 1 
 1 
  
1  
1  
  1 
 1 
 
 1  
1  
 
1   1 
 
 
 1  
 
 1 


center Value
Equation  Values for light blue diagonal 
½(n^{2} + 5)  27 
½(n^{2} + 7)  28 
½(n^{2}  5)  22 
½(n^{2}  3)  23 
½(n^{2}  1)  24 
½(n^{2} + 1)  25 
½(n^{2} + 3)  26 

These new Bachet de Méziriac type squares, which I will label
MZK7^{*} (center cell#) (K[L1,L2] or K[L3,L4]) where (MZK7^{*} signifies a true 7x7 MéziriacKnight
square with the center cell number of the square and a break followed by:
 a knight move L1= (½(n  3) cells Down, L2= ½(n  1) cells Right) for the yellow
broken diagonal squares or
 a knight move L3= (½(n  3) cell Left,L4= ½(n  1) cells Up) for the
light blue broken diagonal.
The squares exhibit the following properties:
 Every number on the main diagonal is represented at least once in this type of square.
 No odd squares divisible by 3, i.e., 3(2n + 1) are possible by this method,
although these squaresdo have some partial magic properties.
Construction of Regular and NonRegular MéziriacKnight Squares
5x5 and 7x7 Squares
For the first square MZK5^{*} 13 K[1L,2U]:
 In the first row place the number 1 one cell to the right of the center column of a 7x7 square and fill in cells by advancing diagonally upwards to the right
until blocked by a previous number.
 Move one cell to the left and two cells up.
 Repeat the process until the square is filled, as shown below in squares 13.


⇒ 
2
2  19  6 
 15 
18  10  
14  1 
9   13 
5  17 
 12  4 
16  8 
11  3  
7  

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

Note that this squares is regular and complementary. The squares that follow are all nonregular and not complementary and just the final squares without
their construction are given.

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


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


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


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

The complementary tables for the last four 5x5 squares may be seen on new Bachet de Méziriac squares,
new Loubère squares and
new Bachet de Méziriac squares. Note that both de Méziriac and Loubère squares can give rise to the
same complementary table.
For the first square MZK7^{*} 25 K[2L,3U]:

1
6   11 
 
 
 10  
 
 5 
9   
 
4  
  
 3 
 8 
  
2  
14  
  1 
 13 
 
 7  
12  
 

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

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

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

Note that this squares is regular and complementary. The following three 7x7 squares (out of a possible 7) are all nonregular and not complementary
and just the final squares are given.
MZK7^{*} 28 LEFT[2L,3U]
2  29  14 
41  19 
46  24 
35  13  40 
18  45 
23  1 
12  39  17 
44  22 
7  34 
38  16  43 
28  6 
33  11 
15  49  27 
5  32 
10  37 
48  26  4 
31  9 
36  21 
25  3  30 
8  42 
20  47 


MZK7^{*} 24 LEFT[2L,3U]
5  32  10 
37  15 
49  27 
31  9  36 
21  48 
26  4 
8  42  20 
47  25 
3  30 
41  19  46 
24  2 
29  14 
18  45  23 
1  35 
13  40 
44  22  7 
34  12 
39  17 
28  6  33 
11  38 
16  43 


MZK7^{*} 27 LEFT[2L,3U]
1  35  13 
40  18 
45  23 
34  12  39 
17  44 
22  7 
11  38  16 
43  28 
6  33 
37  15  49 
27  5 
32  10 
21  48  26 
4  31 
9  36 
47  25  3 
30  8 
42  20 
24  2  29 
14  41 
19  46 

The complementary tables for the last three 7x7 squares may be seen on new bachet squares (Part II) and
new bachet squares (Part III).
An Example of an 11x11 Square
The following is an example of a nonregular 11x11 square from the broken blue diagonal and
shows the[4,5] knight moves at the 1^{st} and 7^{th} breaks:
MZK11^{*} 58 K[4L,5U]
6  70  13 
88  31  95 
38  102  45 
120  63 
69  12  87 
30  94  37 
101  55  119 
62  5 
22  86  29 
93  36  100 
54  118  61 
4  68 
85  28  92 
35  110  53 
117  60  3 
67  21 
27  91  34 
109  52  116 
59  2  77 
20  84 
90  44  108 
51  115  58 
1  76  19 
83  26 
43  107  50 
114  57  11 
75  18  82 
25  89 
106  49  113 
56  10  74 
17  81  24 
99  42 
48  112  66 
9  73  16 
80  23  98 
41  105 
111  65  8 
72  15  79 
33  97  40 
104  47 
64  7  71 
14  78  32 
96  39  103 
46  121 
The Plane of LoubèreKnight 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
MZK7^{*} 28 LEFT[2L,3U] one can move up the right diagonal on a plane and generate
the complete set of 7 squares as is shown in Part IV of the Bachet de Méziriac series.
This completes this section on the new Bachet de Méziriac Knight method and squares (Part I). To continue Méziriac method
(Part II).
To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com