A New Procedure for Magic Squares (Part III)
Méziriac Block Modified Squares
A Discussion of the New Method
An important general principle for generating odd magic squares by the 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 regular Méziriac square is shown below as an example:
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 
Normally the Méziriac method involves a stepwise approach of consecutive numbers, i.e., 1,2,3...n^{2}. For example,
A new generalized Méziriac 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}, the final square is transformed into a modified
Méziriac, having numbers greater than n^{2} by modifying a block or grid of numbers. The initial number is added either at
the normal Méziriac
sites, i.e., (the center of the last row or last column) or on the main diagonal. Every number on the diagonal can be modified, up to n  1.
Replacement of all n numbers on the diagonal leads to semimagic squares.
The generation of these magic squares involves an almost normal approach:
Numbers are added consecutively starting out with 1 and ending with numbers greater
than n^{2} after modification. A break involves
translational moves (up, down or sideways).
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.
Construction of 5x5 and 7x7 Block Modified Méziriac Squares
Method IA: Start at right of center (1, 2 ⇒ break (2))
 Place 1 into the first cell right of center.
 Add the next consecutive number.
 Move, i.e. break, two cells left.
 Repeat the process until the square is filled as shown below in squares 13.
 As shown below this square is not magic because the two columns and two rows (in grey) sum to 50 instead of 75.
 Where the sums 50 cross corresponds to the small square in blue.
 Only the numbers 1 and 2 may be changed to 75.
 Adding n^{2} = 25 to both 1 and 2 (color change to light green) gives
the magic square 3 where S = ½(n^{3} + 5n).

⇒ 
2
 75 
23  10  22 
4  16  75 
9  21  3 
15  2  50 
20  7  14 
1  8  50 
6  13  25 
12  19  75 
17  24  11 
18  5  75 
75  75  75 
50  50  75 

⇒ 
3
 75 
23  10  22 
4  16  75 
9  21  3 
15  27  75 
20  7  14 
26  8  75 
6  13  25 
12  19  75 
17  24  11 
18  5  75 
75  75  75 
75  75  75 

Method II: Start on the main diagonal one cell up from the center cell (1, 2 ⇒ break (2))
 Place 1 one cell up from the center of the main diagonal.
 Add consecutive numbers up to the right hand corner.
 Move, i.e. break, two cells left.
 Repeat the process until the square is filled as shown below in squares 12.
 As shown below this square is not magic because the two columns and two rows (in grey) sum to 50 instead of 75.
 Where the sums 50 cross corresponds to the small square in blue.
 Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
 Adding n^{2} = 25 to both 8 and 15 gives the magic square 3
where S = ½(n^{3} + 5n).
 Starting over and breaking two cells right instead of left gives another magic square 4.

⇒ 
2
 75 
9  21  3 
15  2  50 
20  7  14 
1  8  50 
6  13  25 
12  19  75 
17  24  11 
18  5  75 
23  10  22 
4  16  75 
75  75  75 
50  50  75 

⇒ 
3 Break left
 75 
9  21  3 
40  2  75 
20  7  14 
1  33  75 
6  13  25 
12  19  75 
17  24  11 
18  5  75 
23  10  22 
4  16  75 
75  75  75 
75  75  75 


4 Break right
 75 
15  3  21 
34  2  75 
7  20  8 
1  39  75 
19  12  25 
13  6  75 
11  24  17 
5  18  75 
23  16  4 
22  10  75 
75  75  75 
75  75  75 

Method IB: Start on the main diagonal one cell up from the center cell (1, 2, 3, 4 ⇒ break)
 Place Place 1 one cell up from the center of the main diagonal.
 Add consecutive numbers up to the n  1 cell.
 Move, i.e. break, two cells left.
 Repeat the process until the square is filled as shown below in squares 12.
 As shown below this square is not magic because the two columns and two rows (in grey) sum to 35 and 60 instead of 85.
 Where the sums 60 cross corresponds to the small squares in blue.
 Since the numbers 1, 2, 3 and 4 sit on a diagonal which sums to 35, two of these numbers will be modified.
 Adding n^{2} = 25 to either (3,4,13,15) or (10,17,13,15)
(color change to light green) gives the magic squares 3 and 4 where
where S = ½(n^{3} + 9n).

⇒ 
2
 35 
14  21  8 
15  2  60 
20  7  19 
1  13  60 
6  18  25 
12  24  85 
17  4  11 
23  5  60 
3  10  22 
9  16  60 
60  60  85 
60  60  85 

⇒ 
3
 85 
14  21  8 
40  2  85 
20  7  19 
1  38  85 
6  18  25 
12  24  85 
17  29  11 
23  5  85 
28  10  22 
9  16  85 
85  85  85 
85  85  85 

+ 
4
 85 
14  21  8 
15  27  85 
20  7  19 
26  13  85 
6  18  25 
12  24  85 
42  4  11 
23  5  85 
3  35  22 
9  16  85 
85  85  85 
85  85  85 

Construction of 7x7 Block Modified Méziriac Squares
Method IA: Start at right of center (1, 2, 3 ⇒ break (2))
 Generate the square and its sums.
 As shown below this square is not magic because the three columns and three rows (in grey) sum to 147 instead of 196.
 Where the sums 147 cross corresponds to the small square in blue.
 For example two squares 2 and 3 where the set (1,2,3) or (2,4,42) are modified by the addition of n^{2}
= 49 (color change to light green).
 The magic sum S = ½(n^{3} + 7n).
1
 196 
46  21  45  13 
37  5  29  196 
20  44  12  36 
4  28  3 
147 
43  11  35  10 
27  2  19 
147 
17  34  9  26 
1  18  42 
147 
33  8  25  49 
24  41  16  147 
7  31  48  23 
40  15  32  147 
30  47  22  39 
14  38  6  147 
196  196  196 
196  147  147 
147  147 

⇒ 
2
 196 
46  21  45  13 
37  5  29  196 
20  44  12  36 
4  28  52 
196 
43  11  35  10 
27  51  19 
196 
17  34  9  26 
50  18  42 
196 
33  8  25  49 
24  41  16  196 
7  31  48  23 
40  15  32  196 
30  47  22  39 
14  38  6  196 
196  196  196 
196  196  196 
196  196 

+ 
3
 196 
46  21  45  13 
37  5  29  196 
20  44  12  36 
53  28  3 
196 
43  11  35  10 
27  51  19 
196 
17  34  9  26 
1  18  91 
196 
33  8  25  49 
24  41  16  196 
7  31  48  23 
40  15  32  196 
30  47  22  39 
14  38  6  196 
196  196  196 
196  196  196 
196  196 

Method IB: Start at right of center (1, 2, 3, 4, 5 ⇒ break (2))
 Generate the square and its sums.
 As shown below this square is not magic because the three columns and three rows (in grey) sum to 112 and 161 instead of 210.
 Where the sums 112 and 161 cross corresponds to the small squares in blue.
 For example two squares 2 and 3 where the set (2,3,14,17,21) or (4,5,11,25,26) are modified by the addition of n^{2}
= 49 (color change to light green).
 The magic sum S = ½(n^{3} + 11n).
1
 112 
20  44  19  36 
11  28  3  161 
43  18  35  10 
27  2  26 
161 
17  34  9  33 
1  25  42 
161 
40  8  32  49 
24  41  16 
210 
7  31  48  23 
47  15  39  210 
30  5  22  46 
14  38  6 
161 
4  21  45  13 
37  12  29 
161 
161  161  210 
210  161  161 
161  210 

⇒ 
2
 210 
20  44  19  36 
11  28  51 
210 
43  18  35  10 
27  50  26 
210 
66  34  9  33 
1  25  42 
210 
40  8  32  49 
24  41  16 
210 
7  31  48  23 
47  15  39  210 
30  5  22  46 
63  38  9 
210 
4  70  45  13 
37  12  29 
210 
210  210  210 
210  210  210 
210  210 

+ 
3
 210 
20  44  19  36 
60  28  3 
210 
43  18  35  10 
27  2  75 
210 
17  34  9  33 
1  74  42 
210 
40  8  32  49 
24  41  16 
210 
7  31  48  23 
47  15  39  210 
30  54  22  46 
14  38  9 
210 
53  21  45  13 
37  12  29 
210 
210  210  210 
210  210  210 
210  210 

This completes this section on the new block Méziriac Method (Part III). To return to Part II.
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Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com