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Magic squares (most perfect, [Franklin] panmagic & inlaid)
Detailed explanation about the structure and construction of magic squares
3x extra magic 15x15 square
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How to produce extra magic 15x15 squares?
 
 On the previous web page you can see how to produce a panmagic 15x15 square. On this web page
I show you three different methods to produce extra magic 15x15 squares. These squares are not pan-
magic, but the squares have extra magic features instead.
 

 
Method 1, using a 3x3 and a 5x5 to produce a 15x15 magic square

You can use a magic 3x3 square and a (pan)magic 5x5 square to produce a 15x15 magic square (I have
used this method to produce an [extra] panmagic 35x35 square).
 
 
Make a 5x5 carpet of the magic 3x3 square and make a 3x3 carpet of the (pan)magic 5x5 square. Take
[digit -/- 1] x 25 from the first carpet and add the digit from the same cell of the second capet.
 
 
  25 x [digit -/- 1] from 5x5 carpet of a magic 3x3 square
2
9
4
2
9
4
2
9
4
2
9
4
2
9
4
7
5
3
7
5
3
7
5
3
7
5
3
7
5
3
6
1
8
6
1
8
6
1
8
6
1
8
6
1
8
2
9
4
2
9
4
2
9
4
2
9
4
2
9
4
7
5
3
7
5
3
7
5
3
7
5
3
7
5
3
6
1
8
6
1
8
6
1
8
6
1
8
6
1
8
2
9
4
2
9
4
2
9
4
2
9
4
2
9
4
7
5
3
7
5
3
7
5
3
7
5
3
7
5
3
6
1
8
6
1
8
6
1
8
6
1
8
6
1
8
2
9
4
2
9
4
2
9
4
2
9
4
2
9
4
7
5
3
7
5
3
7
5
3
7
5
3
7
5
3
6
1
8
6
1
8
6
1
8
6
1
8
6
1
8
2
9
4
2
9
4
2
9
4
2
9
4
2
9
4
7
5
3
7
5
3
7
5
3
7
5
3
7
5
3
6
1
8
6
1
8
6
1
8
6
1
8
6
1
8
 
 
  + 1x digit from 3x3 carpet of a (pan)magic 5x5 square
1
19
7
25
13
1
19
7
25
13
1
19
7
25
13
10
23
11
4
17
10
23
11
4
17
10
23
11
4
17
14
2
20
8
21
14
2
20
8
21
14
2
20
8
21
18
6
24
12
5
18
6
24
12
5
18
6
24
12
5
22
15
3
16
9
22
15
3
16
9
22
15
3
16
9
1
19
7
25
13
1
19
7
25
13
1
19
7
25
13
10
23
11
4
17
10
23
11
4
17
10
23
11
4
17
14
2
20
8
21
14
2
20
8
21
14
2
20
8
21
18
6
24
12
5
18
6
24
12
5
18
6
24
12
5
22
15
3
16
9
22
15
3
16
9
22
15
3
16
9
1
19
7
25
13
1
19
7
25
13
1
19
7
25
13
10
23
11
4
17
10
23
11
4
17
10
23
11
4
17
14
2
20
8
21
14
2
20
8
21
14
2
20
8
21
18
6
24
12
5
18
6
24
12
5
18
6
24
12
5
22
15
3
16
9
22
15
3
16
9
22
15
3
16
9
 
 
  = magic 15x15 square
26
219
82
50
213
76
44
207
100
38
201
94
32
225
88
160
123
61
154
117
60
173
111
54
167
110
73
161
104
67
139
2
195
133
21
189
127
20
183
146
14
177
145
8
196
43
206
99
37
205
93
31
224
87
30
218
81
49
212
80
172
115
53
166
109
72
165
103
66
159
122
65
153
116
59
126
19
182
150
13
176
144
7
200
138
1
194
132
25
188
35
223
86
29
217
85
48
211
79
42
210
98
36
204
92
164
102
70
158
121
64
152
120
58
171
114
52
170
108
71
143
6
199
137
5
193
131
24
187
130
18
181
149
12
180
47
215
78
41
209
97
40
203
91
34
222
90
28
216
84
151
119
57
175
113
51
169
107
75
163
101
69
157
125
63
135
23
186
129
17
185
148
11
179
142
10
198
136
4
192
39
202
95
33
221
89
27
220
83
46
214
77
45
208
96
168
106
74
162
105
68
156
124
62
155
118
56
174
112
55
147
15
178
141
9
197
140
3
191
134
22
190
128
16
184
 
 
Notify that each random chosen 3x5 or 5x3 rectangle gives the magic sum of 1695.
 

 
Method 2, using 9x sequencial 5x5 and structure of 3x3 to produce
15x15 magic square

We produce the 15x15 magic square by using 9 panmagic 5x5 squares. We take one panmagic
5x5 square and add each time 25 to all digits. Now we have got 9 panmagic 5x5 squares with all
the digits from 1 up to 225. We put the 9 panmagic 5x5 squares in order of a magic 3x3 square.
 
 
  9 x panmagic 5x5 squares (by adding each time 25 to all digits)
 
 
1
 
 
 
 
 
2
 
 
 
 
 
3
 
 
1
19
7
25
13
 
26
44
32
50
38
 
51
69
57
75
63
10
23
11
4
17
 
35
48
36
29
42
 
60
73
61
54
67
14
2
20
8
21
 
39
27
45
33
46
 
64
52
70
58
71
18
6
24
12
5
 
43
31
49
37
30
 
68
56
74
62
55
22
15
3
16
9
 
47
40
28
41
34
 
72
65
53
66
59
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4
 
 
 
 
 
5
 
 
 
 
 
6
 
 
76
94
82
100
88
 
101
119
107
125
113
 
126
144
132
150
138
85
98
86
79
92
 
110
123
111
104
117
 
135
148
136
129
142
89
77
95
83
96
 
114
102
120
108
121
 
139
127
145
133
146
93
81
99
87
80
 
118
106
124
112
105
 
143
131
149
137
130
97
90
78
91
84
 
122
115
103
116
109
 
147
140
128
141
134
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
7
 
 
 
 
 
8
 
 
 
 
 
9
 
 
151
169
157
175
163
 
176
194
182
200
188
 
201
219
207
225
213
160
173
161
154
167
 
185
198
186
179
192
 
210
223
211
204
217
164
152
170
158
171
 
189
177
195
183
196
 
214
202
220
208
221
168
156
174
162
155
 
193
181
199
187
180
 
218
206
224
212
205
172
165
153
166
159
 
197
190
178
191
184
 
222
215
203
216
209
 
 
  Put the 5x5 squares in order of a 3x3 square
2
2
2
2
2
9
9
9
9
9
4
4
4
4
4
2
2
2
2
2
9
9
9
9
9
4
4
4
4
4
2
2
2
2
2
9
9
9
9
9
4
4
4
4
4
2
2
2
2
2
9
9
9
9
9
4
4
4
4
4
2
2
2
2
2
9
9
9
9
9
4
4
4
4
4
7
7
7
7
7
5
5
5
5
5
3
3
3
3
3
7
7
7
7
7
5
5
5
5
5
3
3
3
3
3
7
7
7
7
7
5
5
5
5
5
3
3
3
3
3
7
7
7
7
7
5
5
5
5
5
3
3
3
3
3
7
7
7
7
7
5
5
5
5
5
3
3
3
3
3
6
6
6
6
6
1
1
1
1
1
8
8
8
8
8
6
6
6
6
6
1
1
1
1
1
8
8
8
8
8
6
6
6
6
6
1
1
1
1
1
8
8
8
8
8