Perfect magic squaresContact / guestbook3x3 magic square3x3 magic square, explanationSudoku method (1)Sudoku method (2)Sudoku method (3)Pan magic 4x4 squarePan magic 4x4 square, explanationTransformation methodPan magic 5x5 squarePan magic 5x5 square, explanation6x6 magic squareKhajuraho methodKhajuraho method, explanationBasic pattern method (1)Basic pattern method (2)Basic pattern method (3)Analysis Franklin panm. 8x8Basic key method (1)Basic key method (2)pan magic 9x9 squarePan magic 15x15 squarePan magic 27x27 squareBordered squaresEach magic sumWater retention challengeFavorite Links
Perfect magic squares
Basic pattern method (1)
From panmagic to Franklin panmagic (basic pattern method of construction)
 
 
From each (pattern of a) 4x4 pan magic square you can produce an 8x8 Franklin pan magic square.
The 8x8 square contains the following digits from 1 to 64 (= 4 x 16 digits):
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
 
 
Select a random 4x4 panmagic square and split the square in two subsquares as follows:

 
16
3
10
5
 
 
16
 
 
5
 
 
 
3
10
 
2
13
8
11
 
 
 
13
8
 
 
 
2
 
 
11
7
12
1
14
 
 
 
12
1
 
 
 
7
 
 
14
9
6
15
4
 
 
9
 
 
4
 
 
 
6
15
 
 
 
Enter digits from the same sub-square in the empty cells crosswise:
 
16
4
9
5
 
 
15
3
10
6
1
13
8
12
 
 
2
14
7
11
8
12
1
13
 
 
7
11
2
14
9
5
16
4
 
 
10
6
15
3
 
 
Combine the 2 sub-squares and add the same 2 sub-squares to the bottom:
 
 
16
4
9
5
15
3
10
6
1
13
8
12
2
14
7
11
8
12
1
13
7
11
2
14
9
5
16
4
10
6
15
3
16
4
9
5
15
3
10
6
1
13
8
12
2
14
7
11
8
12
1
13
7
11
2
14
9
5
16
4
10
6
15
3
 
 
Substitute the digits for the digits with the right colour from the abovementioned table:
 
 
16
52
9
53
15
51
10
54
1
61
8
60
2
62
7
59
56
12
49
13
55
11
50
14
57
5
64
4
58
6
63
3
32
36
25
37
31
35
26
38
17
45
24
44
18
46
23
43
40
28
33
29
39
27
34
30
41
21
48
20
42
22
47
19
 
 
In this way you can produce an 8x8 Franklin panmagic square from every 4x4 panmagic square.
 
You can find an easy method of construction to produce larger (than 8x8) Franklin panmagic on the
‘Basic key method (1)’ page.
 
 

Information for whiz kids:


PERFECT 16x16 FRANKLIN PANMAGIC SQUARE

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

 
 Basic (row) pattern 16x16 square
15
3
10
6
16
4
9
5
15
3
10
6
16
4
9
5
2
14
7
11
1
13
8
12
2
14
7
11
1
13
8
12
7
11
2
14
8
12
1
13
7
11
2
14
8
12
1
13
10
6
15
3
9
5
16
4
10
6
15
3
9
5
16
4
15
3
10
6
16
4
9
5
15
3
10
6
16
4
9
5
2
14
7
11
1
13
8
12
2
14
7
11
1
13
8
12
7
11
2
14
8
12
1
13
7
11
2
14
8
12
1
13
10
6
15
3
9
5
16
4
10
6
15
3
9
5
16
4
15
3
10
6
16
4
9
5
15
3
10
6
16
4
9
5
2
14
7
11
1
13
8
12
2
14
7
11
1
13
8
12
7
11
2
14
8
12
1
13
7
11
2
14
8
12
1
13
10
6
15
3
9
5
16
4
10
6
15
3
9
5
16
4
15
3
10
6
16
4
9
5
15
3
10
6
16
4
9
5
2
14
7
11
1
13
8
12
2
14
7
11
1
13
8
12
7
11
2
14
8
12
1
13
7
11
2
14
8
12
1
13
10
6
15
3
9
5
16
4
10
6
15
3
9
5
16
4
 
 
If you place the digits from 1 to 256 in rows of 16 under each other, then you get 16 columns (of 16 digits).
The column pattern of the 16x16 square is as follows:
 
 
 Column pattern 16x16 square
3
16
1
14
3
16
1
14
1
14
3
16
1
14
3
16
1
14
3
16
1
14
3
16
3
16
1
14
3
16
1
14
16
3
14
1
16
3
14
1
14
1
16
3
14
1
16
3
14
1
16
3
14
1
16
3
16
3
14
1
16
3
14
1
4
15
2
13
4
15
2
13
2
13
4
15
2
13
4
15
2
13
4
15
2
13
4
15
4
15
2
13
4
15
2
13
15
4
13
2
15
4
13
2
13
2
15
4
13
2
15
4
13
2
15
4
13
2
15
4
15
4
13
2
15
4
13
2
7
12
5
10
7
12
5
10
5
10
7
12
5
10
7
12
5
10
7
12
5
10
7
12
7
12
5
10
7
12
5
10
12
7
10
5
12
7
10
5
10
5
12
7
10
5
12
7
10
5
12
7
10
5
12
7
12
7
10
5
12
7
10
5
8
11
6
9
8
11
6
9
6
9
8
11
6
9
8
11
6
9
8
11
6
9
8
11
8
11
6
9
8
11
6
9
11
8
9
6
11
8
9
6
9
6
11
8
9
6
11
8
9
6
11
8
9
6
11
8
11
8
9
6
11
8
9
6
 
 
Use the following formula: Add to a cell from the basic pattern the same cell from the column pattern
minus 1 multiplied by 16 (= the size of the square). For example the top cell at the left: 15 + (3-1) x
16 = 47.


PERFECT FRANKLIN PANMAGIC 32x32 SQUARE

It is even possible to enlarge the 16x16 square to a perfect Franklin pan magic 32x32 square
and then to a 64x64 square, and then to a 128x128 square, and then ….
 

 Left half of the column pattern 32x32 square
11
64
1
54
11
64
1
54
9
62
3
56
9
62
3
56
1
54
11
64
1
54
11
64
3
56
9
62
3
56
9
62
64
11
54
1
64
11
54
1
62
9
56
3
62
9
56
3
54
1
64
11
54
1
64
11
56
3
62
9
56
3
62
9
12
63
2
53
12
63
2
53
10
61
4
55
10
61
4
55
2
53
12
63
2
53
12
63
4
55
10
61
4
55
10
61
63
12
53
2
63
12
53
2
61
10
55
4
61
10
55
4
53
2
63
12
53
2
63
12
55
4
61
10
55
4
61
10
15
60
5
50
15
60
5
50
13
58
7
52
13
58
7
52
5
50
15
60
5
50
15
60
7
52
13
58
7
52
13
58
60
15
50
5
60
15
<