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Perfect magic squares
Basic pattern method (3)
Information for whiz kids:
 
 
FRANKLIN PANMAGIC 8x8 SQUARE:
 
On page ‘Basic pattern method (1)’ you can see how to use the patterns of a 4x4 pan magic square to produce
a Franklin pan magic 8x8 square. The pattern of the panmagic 4x4 square needed to split up. But, it is also
possible to use a 2x2 carpet of a (don’t split it up!) panmagic 4x4 square to produce a Franklin panmagic 8x8 square.
You need one of the 384 panmagic 4x4 squares plus a fixed pattern; see below:
 
 
 1x digit (from 2x2 carpet)                        +      16x digit (from fixed pattern)     =       Franklin panmagic 8x8 square
1
8
13
12
1
8
13
12
 
 
2
1
3
0
3
0
2
1
 
 
33
24
61
12
49
8
45
28
15
10
3
6
15
10
3
6
 
 
1
2
0
3
0
3
1
2
 
 
31
42
3
54
15
58
19
38
4
5
16
9
4
5
16
9
 
 
0
3
1
2
1
2
0
3
 
 
4
53
32
41
20
37
16
57
14
11
2
7
14
11
2
7
 
 
3
0
2
1
2
1
3
0
 
 
62
11
34
23
46
27
50
7
1
8
13
12
1
8
13
12
 
 
0
3
1
2
1
2
0
3
 
 
1
56
29
44
17
40
13
60
15
10
3
6
15
10
3
6
 
 
3
0
2
1
2
1
3
0
 
 
63
10
35
22
47
26
51
6
4
5
16
9
4
5
16
9
 
 
2
1
3
0
3
0
2
1
 
 
36
21
64
9
52
5
48
25
14
11
2
7
14
11
2
7
 
 
1
2
0
3
0
3
1
2
 
 
30
43
2
55
14
59
18
39
 


PERFECT FRANKLIN PANMAGIC 16x16 SQUARE:
 
It is possible to use a 2x2 carpet of a Franklin panmagic 8x8 square to produce a perfect Franklin pan magic
16x16 square. You need a Franklin panmagic 8x8 square plus a fixed pattern; see below:
 
 
  1x digit from 2x2 carpet (of a Franklin panmagic 8x8 square)
33
24
61
12
49
8
45
28
33
24
61
12
49
8
45
28
31
42
3
54
15
58
19
38
31
42
3
54
15
58
19
38
4
53
32
41
20
37
16
57
4
53
32
41
20
37
16
57
62
11
34
23
46
27
50
7
62
11
34
23
46
27
50
7
1
56
29
44
17
40
13
60
1
56
29
44
17
40
13
60
63
10
35
22
47
26
51
6
63
10
35
22
47
26
51
6
36
21
64
9
52
5
48
25
36
21
64
9
52
5
48
25
30
43
2
55
14
59
18
39
30
43
2
55
14
59
18
39
33
24
61
12
49
8
45
28
33
24
61
12
49
8
45
28
31
42
3
54
15
58
19
38
31
42
3
54
15
58
19
38
4
53
32
41
20
37
16
57
4
53
32
41
20
37
16
57
62
11
34
23
46
27
50
7
62
11
34
23
46
27
50
7
1
56
29
44
17
40
13
60
1
56
29
44
17
40
13
60
63
10
35
22
47
26
51
6
63
10
35
22
47
26
51
6
36
21
64
9
52
5
48
25
36
21
64
9
52
5
48
25
30
43
2
55
14
59
18
39
30
43
2
55
14
59
18
39
 
 
  + 64x digit from the fixed pattern
2
1
3
0
3
0
2
1
3
0
2
1
2
1
3
0
1
2
0
3
0
3
1
2
0
3
1
2
1
2
0
3
0
3
1
2
1
2
0
3
1
2
0
3
0
3
1
2
3
0
2
1
2
1
3
0
2
1
3
0
3
0
2
1
0
3
1
2
1
2
0
3
1
2
0
3
0
3
1
2
3
0
2
1
2
1
3
0
2
1
3
0
3
0
2
1
2
1
3
0
3
0
2
1
3
0
2
1
2
1
3
0
1
2
0
3
0
3
1
2
0
3
1
2
1
2
0
3
0
3
1
2
1
2
0
3
1
2
0
3
0
3
1
2
3
0
2
1
2
1
3
0
2
1
3
0
3
0
2
1
2
1
3
0
3
0
2
1
3
0
2
1
2
1
3
0
1
2
0
3
0
3
1
2
0
3
1
2
1
2
0
3
2
1
3
0
3
0
2
1
3
0
2
1
2
1
3
0
1
2
0
3
0
3
1
2
0
3
1
2
1
2
0
3
0
3
1
2
1
2
0
3
1
2
0
3
0
3
1
2
3
0
2
1
2
1
3
0
2
1
3
0
3
0
2
1
 
 
  =  Perfect Franklin panmagic 16x16 square
161
88
253
12
241
8
173
92
225
24
189
76
177
72
237
28
95
170
3
246
15
250
83
166
31
234
67
182
79
186
19
230
4
245
96
169
84
165
16
249
68
181
32
233
20
229
80
185
254
11
162
87
174
91
242
7
190
75
226
23
238
27
178
71
1
248
93
172
81
168
13
252
65
184
29
236
17
232
77
188
255
10
163
86
175
90
243
6
191
74
227
22
239
26
179
70
164
85
256
9
244
5
176
89
228
21
192
73
180
69
240
25
94
171
2
247
14
251
82
167
30
235
66
183
78
187
18
231
33
216
125
140
113
136
45
220
97
152
61
204
49
200
109
156
223
42
131
118
143
122
211
38
159
106
195
54
207
58
147
102
132
117
224
41
212
37
144
121
196
53
160
105
148
101
208
57
126
139
34
215
46
219
114
135
62
203
98
151
110
155
50
199
129
120
221
44
209
40
141
124
193
56
157
108
145
104
205
60
127
138
35
214
47
218
115
134
63
202
99
150
111
154
51
198
36
213
128
137
116
133
48
217
100
149
64
201
52
197
112
153
222
43
130
119
142
123
210
39
158
107
194
55
206
59
146
103
 
 
 
Notify that it is possible to use a 2x2 carpet of a perfect Franklin panmagic 16x16 square to produce a perfect Franklin
panmagic 32x32 square, ...

The fixed pattern is based on the following 4x4 square:
 
 
2
1
3
0
1
2
0
3
0
3
1
2
3
0
2
1
 
 
The fixed pattern can each time be multiplied by four as follows. Put next to the (n x n) square a second square by switching
the left and the right half. Put below the first and the second square a third and a fourth square by switching the top and the
bottom half.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Perfect magic squares|Contact / guestbook|3x3 magic square|3x3 magic square, explanation|Sudoku method (1)|Sudoku method (2)|Sudoku method (3)|Pan magic 4x4 square|Pan magic 4x4 square, explanation|Pan magic 5x5 square|Pan magic 5x5 square, explanation|6x6 magic square|Khajuraho method|Khajuraho method, explanation|Basic pattern method (1)|Basic pattern method (2)|Basic pattern method (3)|Analysis Franklin panm. 8x8|Basic key method (1)|Basic key method (2)|pan magic 9x9 square|Pan magic 15x15 square|Pan magic 27x27 square|Bordered squares|Each magic sum|Water retention challenge|Favorite Links