Franklin Squares
Part 4


My Construction Method  

 

         
Order 12
             
1/3 rows
       
A
1
2
3
4
5
6
odd
even
   
must equal 22
       
1
0
10
9
3
4
5
22
13
 
dia
 
       
2
11
1
2
8
7
6
22
 
15
28
diagonal total
                               
must equal 33
         
Order 12
       
B
1
2
3
4
5
6
 
odd
even
 
       
1
0
9
10
3
7
5
22
17
 
dia
 
       
2
11
2
1
8
4
6
22
 
16
33
 
           
       
C
row one square A
       
1
0
9
10
3
7
5
6
4
8
1
2
11
           
       
D
column one square B
       
1
0
9
10
3
7
5
11
2
1
8
4
6
 
           
       
E
column one square B
       
1
0
9
10
3
7
5
4
6
11
2
1
8
 
               
         
12x12 A
 
12x12 B
         
0
9
10
3
7
5
6
4
8
1
2
11
 
0
11
0
11
0
11
0
11
0
11
0
11
         
11
2
1
8
4
6
5
7
3
10
9
0
 
9
2
9
2
9
2
9
2
9
2
9
2
         
0
9
10
3
7
5
6
4
8
1
2
11
 
10
1
10
1
10
1
10
1
10
1
10
1
         
11
2
1
8
4
6
5
7
3
10
9
0
 
3
8
3
8
3
8
3
8
3
8
3
8
         
0
9
10
3
7
5
6
4
8
1
2
11
 
7
4
7
4
7
4
7
4
7
4
7
4
         
11
2
1
8
4
6
5
7
3
10
9
0
 
5
6
5
6
5
6
5
6
5
6
5
6
         
0
9
10
3
7
5
6
4
8
1
2
11
 
4
7
4
7
4
7
4
7
4
7
4
7
         
11
2
1
8
4
6
5
7
3
10
9
0
 
6
5
6
5
6
5
6
5
6
5
6
5
         
0
9
10
3
7
5
6
4
8
1
2
11
 
11
0
11
0
11
0
11
0
11
0
11
0
         
11
2
1
8
4
6
5
7
3
10
9
0
 
2
9
2
9
2
9
2
9
2
9
2
9
         
0
9
10
3
7
5
6
4
8
1
2
11
 
1
10
1
10
1
10
1
10
1
10
1
10
         
11
2
1
8
4
6
5
7
3
10
9
0
 
8
3
8
3
8
3
8
3
8
3
8
3
         
Magic Square
 
         
1
120
121
48
85
72
73
60
97
24
25
144
 
         
142
27
22
99
58
75
70
87
46
123
118
3
 
         
11
110
131
38
95
62
83
50
107
14
35
134
 
         
136
33
16
105
52
81
64
93
40
129
112
9
 
         
8
113
128
41
92
65
80
53
104
17
32
137
 
         
138
31
18
103
54
79
66
91
42
127
114
7
 
         
5
116
125
44
89
68
77
56
101
20
29
140
 
         
139
30
19
102
55
78
67
90
43
126
115
6
 
         
12
109
132
37
96
61
84
49
108
13
36
133
 
         
135
34
15
106
51
82
63
94
39
130
111
10
 
         
2
119
122
47
86
71
74
59
98
23
26
143
 
         
141
28
21
100
57
76
69
88
45
124
117
4
 
                                 
                               
         
Order 20
       1/5 rows must equal 38                
         
1
2
3
4
5
6
7
8
9
10
 
diagonal total must equal 95
       
1
0
18
17
3
4
14
13
7
8
9
38
38
42
 
       
2
19
1
2
16
15
5
6
12
11
10
38
38
 
44
86
 
           
         
Order 20
         
1
2
3
4
5
6
7
8
9
10
 
       
3
3
18
17
0
4
14
13
7
11
9
38
38
48
 
       
4
16
1
2
19
15
5
6
12
8
10
38
38
 
47
95
 
           
         
Row one square A
         
3
18
17
0
4
14
13
7
11
9
10
8
12
6
5
15
19
2
1
16
 
           
         
Column one square B
         
3
18
17
0
4
14
13
7
11
9
8
10
16
1
2
19
15
5
6
12
 

Orders that are odd multiples of four, 12, 20, 28,…, present problems. In order twelve row total is 66. Half row total is 33. In row one numbers in odd columns are in the diagonal and the compliments of the numbers in even columns are in the diagonal. If odd columns plus even columns equal half row total and odd columns plus compliments of even columns also equal half row total then odd columns=even columns=forth row totals. For all orders that are odd multiples of four half row total will always be odd. For this reason I believe squares of these orders can’t have half rows and/or columns, full diagonals, and bent diagonals with proper totals in the same square.

I’ve given up half rows and half columns with half row totals and added third rows and third columns with third row totals. Any row or column can be broken into three equal segments and each segment will equal one third of row total.

Set up the number set in row A1 and row A2 as before. Adjust the pairs as in rows B1 and B2. In row one of square A row B1 is the left half and row B2 in reverse order is the right half as in C1. This places complimentary pairs symmetrically opposite in the row. In column one of square B complimentary pairs must be both in odd rows or both in even rows. Row D1 shows B1 as the left half and B2 in order as the right half with complimentary pairs in rows one and seven, two and eight, three and nine, and so forth. This places complimentary pairs properly, both elements in odd rows and both elements in even rows, but ruins third column totals. Moving the four and six from rows eleven and twelve to rows seven and eight solve the problem. Row E1 is column one of square B.

Multiply square A by twelve and add square B plus one to get the magic square. It has all the features as the square with symmetry.

The work up for order twenty is shown. In order twenty, rows and columns must be broken into five equal segments with 1/5th row totals.

 

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