A New Procedure for Magic Squares (Part VIII) Continuation
Méziriac Knight Block nontrans and trans Modified 7x7 Squares
A Discussion of the New Method
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 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 7x7 regular Méziriac square is shown below :
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 
Normally the Méziriac methods involve a stepwise approach of consecutive numbers, i.e., 1,2,3...n^{2}.
For example, A new generalized Loubère procedure where the initial number is placed anywhere on any
n  1 cells of the first row.
However, in this new method although the numbers are added consecutively starting with 1 and ending with
n^{2}, to generate an intermediate nonmagic square the final square is transformed into a modified
Méziriac, having numbers greater than
n^{2}.
This is done by taking the nonmagic squares
and converting them into magic ones using transposition
(trans) and nontransposition routes. These squares are shown below in methods I and II.
Accordingly, the numbers are added consecutively starting out with 1 and ending with numbers greater than n^{2}. Whenever a break
occurs a knight move is performed instead of a normal translational move (up, down or sideways).
These square types were covered in Méziriac Knight square methods.
Moreover, it must be stated here that the magic sum has been modified from the known equation
S = ½(n^{3} + n) to
the general equation:
S = ½(n^{3} ± an)
which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or 
values. For example when a = 1 the normal magic sum S is implied.
When a takes on different odd values S gives the magic sum of a modified magic square.
It will be shown that the addition or subtraction of n^{2} to some of the cells in
the square gives rise to a new magic square.
nontransConversion of de Méziriac nonmagic Squares
to Magic Squares
Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
 Begin generating the nonmagic de Méziriac square by placing 1 to the right of the center cell and performing a (1,2) up knight
break.
 Add consecutive numbers and repeat until the square is filled. This produces the nonmagic square 2.
 As shown square 2 is nonmagic since the last column corresponds to how much must be added or subtracted to the previous row numbers to get to 182.

⇒ 
2
 280  
46  4  11  18 
25  32  39  175  +7 
3  10  17  24 
31  38  45  168  +14 
9  16  23  30 
37  44  2  161  +21 
22  29  36  43 
1  18  15  154  +28 
28  35  42  49 
7  14  21  196  14 
34  41  48  6 
13  20  17  189  7 
40  47  5  12 
19  26  33  182  0 
182  182  182 
182  133  182 
182  182  

⇒ 
 Add 49 to 1 to convert all sums in the last row to 182. This also changes 154 to 203.
 The next series of moves (in groups of 2), adding or subtracting numbers from a row converts all the sums to 182 with the generation of duplicate numbers
in green, except for 1 which has no duplicate.
 In order to remove these duplicates add 2n^{2} = 98 to each row
(each duplicate must be included in the modification) so that each row, column and the left diagonal have exactly one modification, i.e, numbers in
pink .
 Square 5 is produced whereby all the sums have been converted to the magic sum 280, and
S = ½(n^{3} + 31n).
3
 280  
46  4  11  18 
25  32  39  175  +7 
3  10  17  24 
31  38  45  168  +14 
9  16  23  30 
37  44  2  161  +21 
22  29  36  43 
50  18  15  203  21 
28  35  42  49 
7  14  21  196  14 
34  41  48  6 
13  20  17  189  7 
40  47  5  12 
19  26  33  182  0 
182  182  182 
182  182  182 
182  182  

⇒ 
4
 280 
46  4  11  18 
32  32  39  182 
3  10  17  38 
31  38  45  182 
30  16  23  30 
37  44  2  182 
1  29  36  43 
50  18  15  182 
28  35  42  35 
7  14  21  182 
34  41  48  6 
6  20  17  182 
40  47  5  12 
19  26  33  182 
182  182  182 
182  182  182 
182  182 

⇒ 
5
 280 
46  4  11  18 
32  130  39  280 
3  10  17  136 
31  38  45  280 
128  16  23  30 
37  44  2  280 
1  29  134  43 
50  18  15  280 
28  133  42  35 
7  14  21  280 
34  41  48  6 
104  20  17  280 
40  47  5  12 
19  26  131  280 
280  280  280 
280  280  280 
280  280 

transConversion of de Méziriac nonmagic Squares to
Magic Squares
Method II: Start at first row center (1 ⇒ (2,1) down knight break)
 Begin generating the nonmagic de Méziriac square by placing 1 to the right of the center cell and performing a (1,2) up knight
break.
 Add consecutive numbers and repeat until the square is filled. This produces the nonmagic square 2.
 As shown square 2 is nonmagic since the last row corresponds to how much must be added or subtracted to the previous column numbers to get to 182.

⇒ 
2
 35 
46  40  34  28 
22  9  3  182 
39  33  27  21 
15  2  45  182 
32  26  20  14 
8  44  38  182 
25  19  13  7 
1  37  31  133 
18  12  6  49 
43  30  24  182 
11  5  48  42 
36  23  17  182 
4  47  41  35 
29  16  10  182 
175  182  189 
196  154  161 
168  182 
+7  0  7  14 
+28  +21  +14  

⇒ 
 Add 49 to 1 to convert all sums in the last column to 182. This also changes 154 to 203.
 To begin converting this square into a magic one, add 21 to each of the entries of the right diagonal (in blue).
 At this point (square 4) all the sums have changed by 21.
3
 35 
46  40  34  28 
22  9  3  182 
39  33  27  21 
15  2  45  182 
32  26  20  14 
8  44  38  182 
25  19  13  7 
50  37  31  182 
18  12  6  49 
43  30  24  182 
11  5  48  42 
36  23  17  182 
4  47  41  35 
29  16  10  182 
175  182  189 
196  203  161 
168  182 
+7  0  7  14 
21  +21  +14  

⇒ 
4
 182 
46  40  34  28 
22  9  24  203 
39  33  27  21 
15  25  45  203 
32  26  20  14 
29  44  38  203 
25  19  13  28 
50  37  31  203 
18  12  27  49 
43  30  24  203 
11  26  48  42 
36  23  17  203 
25  47  41  35 
29  16  10  203 
196  203  210 
217  224  182 
189  203 
+7  0  7  14 
21  +21  +14  

⇒ 
 Transpose (trans), three cells up, all the 7 numbers from the blue
diagonal of square 3 to the blue diagonal of square 5.
At this point the highest sum possible may be 182. This sum may change.
 The next series of moves (in groups of 2), adding or subtracting numbers from a column converts all the sums to 182 with the generation of duplicate numbers
in green (square 6).
 Adding n^{2} = 49 to (6, 11, 17, 29, 30, 33 and 35),
shown in pink , removes five duplicates and gives the magic square 7,
where the magic sum is 231 and where S = ½(n^{3} + 17n).
5
 182 
46  40  34  7 
22  9  24  182 
39  33  6  21 
15  23  45  182 
32  5  20  14 
29  44  38  182 
4  19  13  28 
50  37  31  182 
18  12  27  49 
43  30  3  182 
11  26  48  42 
36  2  17  182 
25  47  41  35 
8  16  10  182 
175  182  189 
196  203  161 
168  182 
+7  0  7  14 
21  +21  +14  

⇒ 
6
 182 
46  40  34  7 
1  30  24  182 
39  33  6  21 
15  23  45  182 
32  5  20  14 
29  44  38  182 
11  19  6  28 
50  37  31  182 
18  12  27  35 
43  30  17  182 
11  26  48  42 
36  2  17  182 
25  47  41  35 
8  16  10  182 
182  182  182 
182  182  182 
182  182 

⇒ 
7
 231 
46  40  34  7 
1  79  24  231 
39  82  6  21 
15  23  45  231 
32  5  20  14 
78  44  38  231 
11  19  55  28 
50  37  31  231 
18  12  27  35 
43  30  66  231 
60  26  48  42 
36  2  17  231 
25  47  41  84 
8  16  10  231 
231  231  231 
231  231  231 
231  231 

This completes this section on the A New Procedure for Méziriac Magic Squares Continuation 7x7 Squares (Part VIII).
To continue to section IXNew Procedure for Magic Squares. To return to homepage.
Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com