Franklin Squares
Part 3


My Construction Method
 

 
A
1
2
3
4
5
6
7
8
 
1
0
1
2
3
4
5
6
7
 
2
15
14
13
12
11
10
9
8
 
 
B
1
2
3
4
5
6
7
8
odd
even
 
3
0
14
13
3
11
5
6
8
60
30
30
 
4
15
1
2
12
4
10
9
7
60
16x16 A
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
0
14
13
3
11
5
6
8
7
9
10
4
12
2
1
15
 
15
1
2
12
4
10
9
7
8
6
5
11
3
13
14
0
 
Magic Square
 
1
240
209
64
177
96
97
144
113
160
161
80
193
48
17
256
 
242
31
34
207
66
175
146
127
130
111
82
191
50
223
226
15
 
16
225
224
49
192
81
112
129
128
145
176
65
208
33
32
241
 
255
18
47
194
79
162
159
114
143
98
95
178
63
210
239
2
 
3
238
211
62
179
94
99
142
115
158
163
78
195
46
19
254
 
244
29
36
205
68
173
148
125
132
109
84
189
52
221
228
13
 
14
227
222
51
190
83
110
131
126
147
174
67
206
35
30
243
 
253
20
45
196
77
164
157
116
141
100
93
180
61
212
237
4
 
5
236
213
60
181
92
101
140
117
156
165
76
197
44
21
252
 
246
27
38
203
70
171
150
123
134
107
86
187
54
219
230
11
 
12
229
220
53
188
85
108
133
124
149
172
69
204
37
28
245
 
251
22
43
198
75
166
155
118
139
102
91
182
59
214
235
6
 
7
234
215
58
183
90
103
138
119
154
167
74
199
42
23
250
 
248
25
40
201
72
169
152
121
136
105
88
185
56
217
232
9
 
10
231
218
55
186
87
106
135
122
151
170
71
202
39
26
247
 
249
24
41
200
73
168
153
120
137
104
89
184
57
216
233
8
                           
C
1
2
3
4
5
6
7
8
9
10
11
12
1
0
1
2
3
4
5
6
7
8
9
10
11
2
23
22
21
20
19
18
17
16
15
14
13
12
                                 
D
1
2
3
4
5
6
7
8
9
10
11
12
odd
even
3
0
22
21
3
4
18
17
7
8
14
13
 11
138
63
75
4
23
1
2
20
19
5
6
16
15
9
10
12
138
                         
E
1
2
3
4
5
6
7
8
9
10
11
12
odd
even
5
0
22
21
3
4
18
17
7
14
8
13
 11
138
69
69
6
23
1
2
20
19
5
6
16
9
15
10
12
138

 

The other feature of interest in square B was complimentary symmetry. If square A can be constructed with that feature it will be in the combined magic square. To accomplish this complimentary pairs must be placed symmetrically opposite the center line. Enter the number set in two lines as shown and block it off into blocks of four pairs. In the first block invert the middle two pairs. In the second block invert the outside two pairs. In the top row the numbers in the odd columns are in the half diagonal. The numbers in the even columns have their compliments in the diagonal. The numbers in the odd columns must total 30 as do the numbers in the even columns to have both half rows and half diagonals with the proper totals. This construction accomplishes that. Place the top row, B3, in the left half of row one in square A and the bottom row, B4, in the right half in reverse order.

This construction also produces a pandiagonal square with all the Franklin features including wraparound of the bent diagonals.

Complete the square with complimentary construction. Multiply this square by 16 and add square B of the previous square plus one.

This construction is general for orders that are multiples of sixteen. They have an even number of blocks of four pairs which is necessary-pairs of blocks offset each other. Orders that are odd multiples of eight have an odd number of blocks and must be adjusted to get the right totals. The workup for order 24 is shown. Note that the center two pairs are inverted in all blocks and the adjustment is made by reversing columns nine and ten. This is only one of many ways to do this. I don’t have a general construction.

 

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