A New Procedure for Magic Squares (Part IV)
Loubère and Méziriac Knight Block Modified Squares
A Discussion of the New Method
An important general principle for generating odd magic squares by the De La Loubère 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 Loubère 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 Loubère square is shown below on the left and the regular Méziriac on the right :
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 


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 Loubère and 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}, the final square is transformed into a modified
Loubère or Méziriac square, having numbers greater than n^{2}. This is done by modifying a block or grid of numbers on the square.
The initial number is added either at the normal Loubère
sites, 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 same for the Méziriac method.
In this new method 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). The knight move is either a (2,1) or a variable knight move , e.g (3,2),
for Méziriac squares or Loubère squares. Both these square types were covered in Loubère Knight and
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.
Construction of 5x5 Block Knight Modified Loubère and
Méziriac Squares
Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
 Place 1 in the center cell of the first row.
 Add consecutive numbers.
 Knight break, two cells up one cell left.
 Repeat the process until the square is filled as shown below in squares 13.
 As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
 Where the sum 45 crosses corresponds to the small square in blue.
 Add 25 to the number 1. At this point all the numbers except for one sum to 70.
 Adding n^{2} = 25 to (3,4,5,15 and 18) gives the magic square 4.
Note that all these modified numbers fall outside the blue square.
In the previous nonknight Part I method all the modified numbers fall within the blue square.
 The magic sum is S = ½(n^{3} + 13n).

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

⇒ 
3
 20 
17  9  1 
13  5  45 
8  25  12 
4  21  70 
24  16  3 
20  7  70 
15  2  19 
11  23  70 
6  18  10 
22  14  70 
70  70  45 
70  70  70 

⇒ 
4
 95 
17  9  26 
13  30  95 
8  25  12 
29  21  95 
24  16  28 
20  7  95 
40  2  19 
11  23  95 
6  43  10 
22  14  95 
95  95  95 
95  95  95 

Method II: Start at lower left hand corner (1, 2, 3, 4 ⇒ (2,1) down knight break)
 Place 1 into the lower left hand corner.
 Add three consecutive numbers up to the rightmost cell.
 Knight break, two cells down one cell left.
 Repeat the process until the square is filled as shown below in squares 13.
 As shown below this square is not magic because all the columns, rows and diagonals don't sum to 90 the magic sum.
 Where the sums 60 cross corresponds to the small square in blue.
 One way of changing four of the numbers in the block is shown next.
 Adding n^{2} = 25 to both 1, 2, 6 and 21
(color change to light green) gives the square 4.
 To convert the square to the fully magic square 5 add 5 to each of the cells (for a total of 25) in the left diagonal,
where each of the cells lastly modified are tan. Note that some of the modified numbers fall outside the
blue square and that no duplicate numbers are introduced.
 The magic sum is S = ½(n^{3} + 11n).

⇒ 

⇒ 
3
 35 
8  11  19 
22  25  85 
10  18  21 
4  7  60 
17  20  3 
6  14  60 
24  2  5 
13  16  60 
1  9  12 
15  23  60 
60  60  60 
60  85  65 

⇒ 
4
 85 
8  11  19 
22  25  85 
10  18  46 
4  7  85 
17  20  3 
31  14  85 
24  27  5 
13  16  85 
26  9  12 
15  23  85 
85  85  85 
85  85  65 

⇒ 
5
 90 
13  11  19 
22  25  90 
10  23  46 
4  7  90 
17  20  8 
31  14  90 
24  27  5 
18  16  90 
26  9  12 
15  28  90 
90  90  90 
90  90  90 

Method III: Start at lower left hand corner (1, 2, 3, 4 ⇒ (2,1) down knight break)
 Place 1 into the center cell of the last row .
 Add 2 consecutive numbers.
 Knight break, two cells up one cell left
 Repeat the process until the square is filled as shown below in squares 12.
 As shown below this square is not magic because all the columns, rows and diagonals don't all sum to one number and possibly 105 is the magic sum.
 One way to modify is using square 3 which converts all sums except for the main diagonal into 80.
 On one of the parallel diagonals (which includes a cell of the left diagonal) add 25 to each of the entries (square 4).
Note that some of the modified numbers fall outside the blue square.
 The magic square sum for square 4 is
S = ½(n^{3} + 17n).

⇒ 
2
 105 
13  25  17 
4  21  80 
24  16  8 
20  12  80 
15  7  19 
11  3  55 
6  23  10 
2  14  55 
22  9  1 
18  5  55 
80  80  55 
55  55  55 

⇒ 
3
 105 
13  25  17 
4  21  80 
24  16  8 
20  12  80 
15  7  19 
11  28  80 
6  23  10 
27  14  80 
22  9  26 
18  5  80 
80  80  80 
80  80  80 

⇒ 
4
 105 
13  25  42 
4  21  105 
24  41  8 
20  12  105 
40  7  19 
11  28  105 
6  23  10 
27  39  105 
22  9  26 
43  5  105 
105  105  105 
105  105  105 

Method IV:Start one cell right of center (1 ⇒ (2,1) up knight break)
Interconversion of a Méziriac Knight Square to a
Loubère Knight Square
 Start out with a Méziriac type square by placing 1 one cell to the right of center.
 Knight break, two cells up one cell left.
 Repeat the process until the square is filled as shown below in squares 12.
 As shown below this square is not yet magic.
 Where the sums 45 cross corresponds to the small square in blue.
 Adding n^{2} = 25 to both (6,7,11,25,26) gives the magic square 4
where S = ½(n^{3} + 13n).
 To convert this Méziriac Knight Square to a Loubère Knight Square 1 is subtracted from every cell to give square 5
where S is now equal to ½(n^{3} + 11n).
 Subtraction of 25 from those cells colored in tan from square 5 gives the Loubère
semimagic Knight square 2b. This square can be generated using the
New De la Loubère Method (Part II), except using a (2,1) up knight break.

⇒ 
2
 45 
23  15  2 
19  11  70 
14  6  18 
10  22  70 
5  17  9 
1  13  45 
21  8  25 
12  4  70 
7  24  16 
3  20  70 
70  70  70 
45  70  70 

⇒ 
3
 45 
23  15  2 
19  11  70 
14  6  18 
10  22  70 
5  17  9 
26  13  70 
21  8  25 
12  4  70 
7  24  16 
3  20  70 
70  70  70 
70  70  70 

⇒ 
4
 95 
23  15  2 
19  36  95 
14  31  18 
10  22  95 
5  17  9 
51  13  95 
21  8  50 
12  4  95 
32  24  16 
3  20  95 
95  95  95 
95  95  95 

⇒ 
5
 90 
22  14  1 
18  35  90 
13  30  17 
9  21  90 
4  16  8 
50  12  90 
20  7  49 
11  3  95 
31  23  15 
2  19  90 
90  90  90 
90  90  90 


⇒ 
2b
 40 
22  14  1 
18  10  65 
13  5  17 
9  21  65 
4  16  8 
25  12  45 
20  7  24 
11  3  65 
6  23  15 
2  19  65 
65  65  65 
45  65  65 

⇒  Square 5 
Method V:Start at right of center (1, 2, 3 ⇒ (2,1) up knight break)
 Place 1 to the right of the center cell.
 For this 5x5 square add consecutive numbers 2 and 3.
 Knight break, two cells up one cell 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 three columns and three rows (in grey) sum to 55 instead of 80.
 Where the sums 55 cross corresponds to the small square in blue.
 Adding n^{2} = 25 to both (1, 2 and3) or (1, 11 and 14) gives the magic squares 3 and 4.
Note that some of the modified numbers fall outside the blue square.
 The magic sum of both squares is S = ½(n^{3} + 7n).

⇒ 
2
 55 
3  15  7 
19  11  55 
14  6  23 
10  2  55 
5  22  9 
1  18  55 
21  13  25 
17  4  80 
12  24  16 
8  20  80 
55  80  80 
55  55  55 

⇒ 
3
 80 
28  15  7 
19  11  80 
14  6  23 
35  2  80 
5  22  9 
1  33  80 
21  13  25 
17  4  80 
12  24  16 
8  20  80 
80  80  80 
80  80  80 

+ 
4
 80 
3  15  7 
19  36  80 
29  6  23 
10  2  80 
5  22  9 
26  18  80 
21  13  25 
17  4  80 
12  24  16 
8  20  80 
80  80  80 
80  80  80 

This completes this section on the new block Loubère Method (Part IV). The next section deals with
Loubère and Méziriac Knight Block Modified 7x7 Squares (Part V). To return to homepage.
Copyright © 2009 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com