New De La Loubère Method and Squares (Part IIIB Continued)
Regular and NonRegular LoubèreVariable Knight Combo Magic and SemiMagic Squares
the Full Monty III
A Discussion of the New Methods
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 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.
the 5x5 and 7x7 regular Loubère squares are shown below as examples:
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 

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

this page which is a continuation of new Loubère Knight methods Part IIIA.
To start we place the number 1 on either of 2 broken diagonals, filling the square
in a Loubère fashion until a block is encountered, then proceeding with a knight move results in the generation of regular and
nonregular squares.
All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the
light grey main diagonal behave differently from typical
Loubère squares. After a break/ knight[2,1] (2 down,1 left) for the broken yellow diagonal
and break/knight[1,2] (1 down, 2 left) for the broken light blue diagonal, one regular square and
n  1
nonregular squares are produced. Squares belonging to one diagonal group are identical to another square on the
other diagonal group.
Using the 5x5 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
(starting with the square generated from 1 in the first row) where the values range from
½(n^{2}  n + 2) to ½(n^{2} + 5).
the set of Broken Diagonals
 
1  1 
 

 1  1 
 
 
1  1 
 

 
1   
 
 1 
 
 
 1  1 
 
 
1  1  
 
 1  1 
 


center Value
Equation  Value K[3U,4R]) 
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 Loubère squares, which I will label
Ln^{*} (center cell#) (K[L1,L2] or K[L3,L4]) where (Ln^{*} signifies a nxn
Loubère square with the center cell number of the square and having a break followed by a Knight move L1= number1(Up) and
L2=number2(Right) Or a Knight move L3= number3(Down) and L4=number4(Left).
this is opposed to the original Loubère squares depicted in the introduction as
L5^{*} 13 [1D] and L7^{*} 25 [1D].
the squares on this page exhibit the following properties:
 Every number on the main diagonal is represented at least once in this type of square.
 For the Knight[3U,4R] method no odd squares divisible by 3 are magic, except for those also divisible by 5 which are semimagic.
 For the Knight[3D,2L] method no odd squares divisible by 3 are magic, however those divisible by 7 are semimagic.
Construction of Regular and Nonregular LoubèreKnight squares
Examples of Magic 7x7 Squares Using the Knight[3U,4R] Method
The result of construction of these squares using the Knight[3U,4R] method is : All 7x7 Knight[3U,4R] squares identical to 7x7
knight[4D,3L] squares.
See bottom of page. Since no Knight[3U,4R] 9x9 square is magic, the 11x11 will be constructed:
An 11x11 Knight[3U,4R] Example
 Place the number 1 at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked
by a previous number.
 Using a knight move 3 up and 4 right.
 Repeat the process until the square is filled, as shown below in squares 14 (Show only first knight move per square).
 Compare to the knight move 4 down and 3
left from New De La Loubère Method and Squares (Part IIIA) which this is meant to complement.
1
  
  1 
21  30  39 
 
  
 11  20 
29  38  
 
  
10  19  28 
37   
 
  9 
18  27  36 
  
 
 8  17 
26  35  
  
 
7  16  25 
34   
  
 
15  24  44 
  
  
 6 
23  43  
  
  
5  14 
42   
  
  4 
13  33 
  
  
 3  12 
32  41 
  
  
2  22  31 
40  

⇒ 
2
  
  1 
21  30  39 
48  57 
  
 11  20 
29  38  47 
56  
  
10  19  28 
37  46  66 
 
  9 
18  27  36 
45  65  
 
 8  17 
26  35  55 
64   
 
7  16  25 
34  54  63 
  
 
15  24  44 
53  62  
  
 6 
23  43  52 
61   
  
5  14 
42  51  60 
  
  4 
13  33 
50  59  
  
 3  12 
32  41 
58  67  
  
2  22  31 
40  49 

⇒ 
3 L11^{*} 63 K[3U,4R]
77  86  95 
104  113  1 
21  30  39 
48  57 
85  94  103 
112  11  20 
29  38  47 
56  76 
93  102  111 
10  19  28 
37  46  66 
75  84 
101  121  9 
18  27  36 
45  65  74 
83  92 
120  8  17 
26  35  55 
64  73  82 
91  100 
7  16  25 
34  54  63 
72  81  90 
110  119 
15  24  44 
53  62  71 
80  89  109 
118  6 
23  43  52 
61  70  79 
99  108  117 
5  14 
42  51  60 
69  78  98 
107  116  4 
13  33 
50  59  68 
88  97  106 
115  3  12 
32  41 
58  67  87 
96  105  114 
2  22  31 
40  49 


A L11^{*} 65 K[4D,3L]
75  80  96 
101  117  1 
17  33  38 
54  59 
79  95  100 
116  11  16 
32  37  53 
58  74 
94  110  115 
10  15  31 
36  52  57 
73  78 
109  114  9 
14  30  35 
51  56  72 
88  93 
113  8  13 
29  34  50 
66  71  87 
92  108 
7  12  28 
44  49  65 
70  86  91 
107  112 
22  27  43 
48  64  69 
85  90  106 
111  6 
26  42  47 
63  68  84 
89  105  121 
5  21 
81  46  62 
67  83  99 
104  120  4 
20  25 
49  61  77 
82  98  103 
119  3  19 
24  40 
60  76  81 
97  102  118 
2  18  23 
39  55 

Note that both squares are nonregular, i.e, not complementary as shown in the connectivities of the complementary table. Also to decrease crowding only the first and middle
eleven entries of the connectivity table are shown, since the second to fifth groups have identical connectivities as the first.
The Knight[3D,2L] Method: A SemiMagic 7x7 Square and 11x11 Magic Square
A 7x7 and 11x11 Knight[3D,2L] Examples which complement the Knight[2U,3R] squares. the 7x7 is the first instance of a semimagic,
wherein the left main diagonal sums to 168. the sums (S+d) of the set of 7x7 squares are also shown, where only one square is fully magic:
A L7^{*} 24 K[3D,2L]]
31  42  46 
1  12 
16  27 
41  45  7 
11  15 
26  30 
44  6  10 
21  25 
29  40 
5  9  20 
24  35 
39  43 
8  19  23 
34  38 
49  4 
18  22  33 
37  48 
3  14 
28  32  36 
47  2 
13  17 


7x7 Square Values of K[3D,2L]
center Value  S + d  d 
22  154  21 
23  161  14 
24  168  7 
25  175  0 
26  182  7 
27  189  14 
28  196  21 


B L11^{*} 60 K[3D,2L]]
69  84  99 
103  118  1 
16  31  35 
50  65 
83  98  102 
117  11  15 
30  34  49 
64  68 
97  101  116 
10  14  29 
44  48  63 
67  82 
100  115  9 
13  28  43 
47  62  77 
81  96 
114  8  12 
27  42  46 
61  76  80 
95  110 
7  22  26 
41  45  60 
75  79  94 
109  113 
21  25  40 
55  59  74 
78  93  108 
112  6 
24  39  54 
58  73  88 
92  107  111 
5  20 
38  53  57 
72  87  91 
106  121  4 
19  23 
52  56  71 
86  90  105 
120  3  18 
33  37 
66  70  85 
89  104  119 
2  17  32 
36  51 

It was mentioned above that All 7x7 Knight[3U,4R] squares identical to 7x7 knight[4D,3L] squares. At this point this applies to all squares which come
under the general formula Knight[(number1*U),(number2*R)] and Knight[(number2*D,number1*L)]. thus for example when n = 11, K[5U,6R] is identical to K[6D,5L]
where number1 = ½(n  1) and number2 = ½(n + 1) and where number1 + number2 = n. thus the same result is obtained by moving in any
of two knight directions. To go back up.
L7^{*} 28 K[3U,4R] or
K[4D,3L]
29  41  46 
2  14 
19  24 
40  45  1 
13  18 
23  35 
44  7  12 
17  22 
34  39 
6  11  16 
28  33 
38  43 
10  15  27 
32  37 
49  5 
21  26  31 
36  48 
4  9 
25  30  42 
47  3 
8  20 
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
L11^{*} 63 K[3U,4R] one can move up the right diagonal on a plane
of four L11^{*} 63 K[3U,4R] 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 LoubèreKnight squares. To go back to previous page
New De La Loubère Method and Squares (Part IIIA).
To return to homepage.
Copyright © 2008 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com