|
A perfect Franklin panmagic square (for each multiple of 8) can be produced using the basic key method of con-
struction. See the basic key method of construction executed to produce a perfect 16x16 Franklin pan
magic square below.
|
1
|
2
|
|
|
3
|
4
|
|
|
5
|
6
|
|
|
7
|
8
|
|
|
|
|
|
1
|
2
|
|
|
3
|
4
|
|
|
5
|
6
|
|
|
7
|
8
|
Ensure that the sum of each column is the size of the magic square plus 1 (so 16 + 1 = 17).
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
Copy the two rows to the bottom of the square until the size of the magic square has been reached.
Column:
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
|
1
|
2
|
16
|
15
|
3
|
4
|
14
|
13
|
5
|
6
|
12
|
11
|
7
|
8
|
10
|
9
|
|
16
|
15
|
1
|
2
|
14
|
13
|
3
|
4
|
12
|
11
|
5
|
6
|
10
|
9
|
7
|
8
|
Produce a second square by rotating the first square (a quarter turn to the left).
Row:
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
8
|
9
|
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
7
|
10
|
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
6
|
11
|
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
5
|
12
|
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
4
|
13
|
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
3
|
14
|
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
2
|
15
|
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
1
|
16
|
Take a digit (= column coordinate) from a cell of the column square and take a digit (= row coordinate) from the same
cell of the row square and look up in the table below the digit you need to fill in.
|
|
|
Column:
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Row:
|
1
|
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
|
|
2
|
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
31
|
32
|
|
|
3
|
|
33
|
34
|
35
|
36
|
37
|
38
|
39
|
40
|
41
|
42
|
43
|
44
|
45
|
46
|
47
|
48
|
|
|
4
|
|
49
|
50
|
51
|
52
|
53
|
54
|
55
|
56
|
57
|
58
|
59
|
60
|
61
|
62
|
63
|
64
|
|
|
5
|
|
65
|
66
|
67
|
68
|
69
|
70
|
71
|
72
|
73
|
74
|
75
|
76
|
77
|
78
|
79
|
80
|
|
|
6
|
|
81
|
82
|
83
|
84
|
85
|
86
|
87
|
88
|
89
|
90
|
91
|
92
|
93
|
94
|
95
|
96
|
|
|
7
|
|
97
|
98
|
99
|
100
|
101
|
102
|
103
|
104
|
105
|
106
|
107
|
108
|
109
|
110
|
111
|
112
|
|
|
8
|
|
113
|
114
|
115
|
116
|
117
|
118
|
119
|
120
|
121
|
122
|
123
|
124
|
125
|
126
|
127
|
128
|
|
|
9
|
|
129
|
130
|
131
|
132
|
133
|
134
|
135
|
136
|
137
|
138
|
139
|
140
|
141
|
142
|
143
|
144
|
|
|
10
|
|
145
|
146
|
147
|
148
|
149
|
150
|
151
|
152
|
153
|
154
|
155
|
156
|
157
|
158
|
159
|
160
|
|
|
11
|
|
161
|
162
|
163
|
164
|
165
|
166
|
167
|
168
|
169
|
170
|
171
|
172
|
173
|
174
|
175
|
176
|
|
|
12
|
|
177
|
178
|
179
|
180
|
181
|
182
|
183
|
184
|
185
|
186
|
187
|
188
|
189
|
190
|
191
|
192
|
|
|
13
|
|
193
|
194
|
195
|
196
|
197
|
198
|
199
|
200
|
201
|
202
|
203
|
204
|
205
|
206
|
207
|
208
|
|
|
14
|
|
209
|
210
|
211
|
212
|
213
|
214
|
215
|
216
|
217
|
218
|
219
|
220
|
221
|
222
|
223
|
224
|
|
|
15
|
|
225
|
226
|
227
|
228
|
229
|
230
|
231
|
232
|
233
|
234
|
235
|
236
|
237
|
238
|
239
|
240
|
|
|
16
|
|
241
|
242
|
243
|
244
|
245
|
246
|
247
|
248
|
249
|
250
|
251
|
252
|
253
|
254
|
255
|
256
|
Result:
|
129
|
114
|
144
|
127
|
131
|
116
|
142
|
125
|
133
|
118
|
140
|
123
|
135
|
120
|
138
|
121
|
|
160
|
111
|
145
|
98
|
158
|
109
|
147
|
100
|
156
|
107
|
149
|
102
|
154
|
105
|
151
|
104
|
|
113
|
130
|
128
|
143
|
115
|
132
|
126
|
141
|
117
|
134
|
124
|
139
|
119
|
136
|
122
|
137
|
|
112
|
159
|
97
|
146
|
110
|
157
|
99
|
148
|
108
|
155
|
101
|
150
|
106
|
153
|
103
|
152
|
|
161
|
82
|
176
|
95
|
163
|
84
|
174
|
93
|
165
|
86
|
172
|
91
|
167
|
88
|
170
|
89
|
|
192
|
79
|
177
|
66
|
190
|
77
|
179
|
68
|
188
|
75
|
181
|
70
|
186
|
73
|
183
|
72
|
|
81
|
162
|
96
|
175
|
83
|
164
|
94
|
173
|
85
|
166
|
92
|
171
|
87
|
168
|
90
|
169
|
|
80
|
191
|
65
|
178
|
78
|
189
|
67
|
180
|
76
|
187
|
69
|
182
|
74
|
185
|
71
|
184
|
|
193
|
50
|
208
|
63
|
195
|
52
|
206
|
61
|
197
|
54
|
204
|
59
|
199
|
56
|
202
|
57
|
|
224
|
47
|
209
|
34
|
222
|
45
|
211
|
36
|
220
|
43
|
213
|
38
|
218
|
41
|
215
|
40
|
|
49
|
194
|
64
|
207
|
51
|
196
|
62
|
205
|
53
|
198
|
60
|
203
|
55
|
200
|
58
|
201
|
|
48
|
223
|
33
|
210
|
46
|
221
|
35
|
212
|
44
|
219
|
37
|
214
|
42
|
217
|
39
|
216
|
|
225
|
18
|
240
|
31
|
227
|
20
|
238
|
29
|
229
|
22
|
236
|
27
|
231
|
24
|
234
|
25
|
|
256
|
15
|
241
|
2
|
254
|
13
|
243
|
4
|
252
|
11
|
245
|
6
|
250
|
9
|
247
|
8
|
|
17
|
226
|
32
|
239
|
19
|
228
|
30
|
237
|
21
|
230
|
28
|
235
|
23
|
232
|
26
|
233
|
|
16
|
255
|
1
|
242
|
14
|
253
|
3
|
244
|
12
|
251
|
5
|
246
|
10
|
249
|
7
|
248
|
The sum of the digits of each quarter row, each quarter column and each quarter diagonal is a quarter of the
magic sum. Also the sum of the digits from all types of circles and other symmetric figures is a proportional
part of the magic sum. The conditions are that each horizontal, vertical and diagonal line of the figure must
consist of an even number of cells and the whole figure must consist of a multiple of 4 cells. Find the sym-
metric figures. Aale de Winkel found the whole magic alphabet; see article "Compact" on website:
www.magichypercubes.com/Encyclopedia
Using the basic key method of construction try to produce a perfect 24x24 or 32x32 Franklin pan-
magic square. Establish that in the 24x24 Franklin panmagic square the sum of the digits of each
1/6, 1/3 and 1/2 of the rows, columns and diagonals is 1/6, 1/3 respectively 1/2 of the magic sum
and in the 32x32 Franklin panmagic square the sum of the digits of each 1/8, 1/4 and 1/2 of the
rows, columns and diagonals is 1/8, 1/4 respectively 1/2 of the magic sum.
See more information on page: Basic key method (2)
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