New De Méziriac Knightstep Method and Squares (Part III)
A Discussion of the New Methods
An important general principle for generating odd magic squares by the 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 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 5x5 and 7x7 regular Méziriac squares are shown below as examples:

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 


4  29  12 
37  20 
45  28 
35  11  36 
19  44 
27  3 
10  42  18 
43  26 
2  34 
41  17  49 
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 

Méziriac squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time.
In this new method each subsequent number is added in a stepwise knight step manner, followed by a right or an up break.
In addition, n odd squares may be constructed with the initial number 1 on either of two broken diagonals shown below in
light blue or yellow separated by symmetry by a
light grey diagonal for a 5x5 square.
The set of 5x5 Méziriac Broken Diagonals
1  
 1 

  1 
 1 
 1  
1  
1   1 
 
 1  
 1 


The set of 7x7 Méziriac Broken Diagonals
1  

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

These new Méziriac type squares, have a two cell break like the regular
Méziriac square and is labeled, as in previous web pages, as follows:
 KMn^{*} (center cell#) [(2U,1L),2L]
where KMn^{*} signifies a Knightstep nxn Méziriac square with the center cell number of the square and breaking
left. These squares are identical to entry 1 above by 180° deg rotation along the horizontal axis.
 KMn^{*} (center cell#) [(1D,2R),2D]
where KMn^{*} signifies a Knightstep nxn Méziriac square with the center cell number of the square and breaking
down.
depending whether the 1's lie on the yellow or blue diagonal.
 Every number on the main diagonal is represented at least once in this type of square.,/li>
 For Méziriac Knightstep where n is divisible by 3, i.e., 3(2n + 1) only
semimagic squares are possible for the (2D,1L),2L or 2D series.
Construction Knightstep Méziriac Magic Squares
The set of 5x5 Squares (Method II)
 To generate the regular square, KM5^{*} 19 [(2D,1L),2L],
place a 1 left of center on a 5x5 square and fill in cells by advancing in a knight fashion to the down to the left until blocked by a previous number.
 Move two cells left.
 Repeat the process until the square is filled, as shown below in squares 15.

⇒ 

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

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

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

The Other Four 5x5 Méziriac Knightstep Squares

A KM5^{*} 22 [(2D,1L),2L]
20  2  14 
21  8 
24  6  18 
5  12 
3  15  22 
9  16 
7  19  1 
13  25 
11  23  10 
17  4 

 
B KM5^{*} 11 [(2D,1L),2L]
9  16  3 
15  22 
13  25  7 
19  1 
17  4  11 
23  10 
21  8  20 
2  14 
5  12  24 
6  18 

 
C KM5^{*} 8 [(2D,1L),2L]
1  13  25 
7  19 
10  17  4 
11  23 
14  21  8 
20  2 
18  5  12 
24  6 
22  9  16 
3  15 

 
D KM5^{*} 5 [(2D,1L),2L]
23  10  17 
4  11 
2  14  21 
8  20 
6  18  5 
12  24 
15  22  9 
16  3 
19  1  13 
25  7 

A 7x7 Méziriac Knightstep Square
The construction of a 7x7 Méziriac Knightstep square KM7^{*} 33 [(2D,1L),2L] from the
broken yellow diagonal is shown below using the knightstep approach.
1
11   3 
 
 
  
8  
7  
 4  
 
 12 
  9 
 1 
 
5   
 
13  
 10  
2  
 
  15 
 14 
 6 

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

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

Three 9x9 Méziriac Knightstep Squares
Three 9x9 Méziriac Knightstep square from the broken yellow diagonal are
shown along with the accompanying table of 9 squares showing the triad sums of the left diagonal.
Square A shows a typical knight (2D,1L) move and left break move.
A KL9^{*} 51 [(2D,1L),2L]
79  29  69 
19  59  18 
49  8  39 
14  54  4 
44  75  34 
65  24  55 
30  70  20 
60  10  50 
9  40  80 
46  5  45 
76  35  66 
25  56  15 
71  21  61 
11  51  1 
41  81  31 
6  37  77 
36  67  26 
57  16  47 
22  62  12 
52  2  42 
73  32  72 
38  78  28 
68  27  58 
17  48  7 
63  13  53 
3  43  74 
33  64  23 


B KL9^{*} 66 [(2D,1L),2L]
13  53  3 
43  74  33 
64  23  63 
29  69  19 
59  18  49 
8  39  79 
54  4  44 
75  34  65 
24  55  14 
70  20  60 
10  50  9 
40  80  30 
5  45  76 
35  66  25 
56  15  46 
21  61  11 
51  1  41 
81  31  71 
37  77  36 
67  26  57 
16  47  6 
62  12  52 
2  42  73 
32  72  22 
78  28  68 
27  58  17 
48  7  38 


C KL9^{*} 9 [(2D,1L),2L]
28  68  27 
58  17  48 
7  38  78 
53  3  43 
74  33  64 
23  63  13 
69  19  59 
18  49  8 
39  79  29 
4  44  75 
34  65  24 
55  14  54 
20  60  10 
50  9  40 
80  30  70 
45  76  35 
66  25  56 
15  46  5 
61  11  51 
1  41  81 
31  71  21 
77  36  67 
26  57  16 
47  6  37 
12  52  2 
42  73  32 
72  22  62 

9x9 Cell Values and S of K[(2D,1L),1R]
center Value  S + d  d 
23  450  81 
68  369  0 
32  288  81 
77  450  81 
41  369  0 
5  288  81 
50  450  81 
14  369  0 
59  288  81 
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
KM7^{*} 33 [(2D,1L),2L] one can move up the right diagonal on a plane of four
KM7^{*} 33 [(2D,1L),2L] 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 regular and nonregular De La Méziriac two step squares (Part I). To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com