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Perfect magic squares
Pan magic 15x15 square
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How to produce pan magic 15x15 squares
 
 
Can you use to produce a 15x15 panmagic square the same method to produce panmagic 5x5 squares? The
answer is yes and no. If you choose as first row the digits 0-1-2-3-4-5-6-7-8-9-10-11-12-13-14 than you get
as result only a semi-magic 15x15 square. If you use the digits 0-2-1-3-4-5-8-7-6-11-10-9-13-12-14, than
you get a correct panmagic 15x15 square.

Notify that the row 0-2-1-3-4-5-8-7-6-11-10-9-13-12-14 leads to a correct result, because 0+3+8+11+13,
2+4+7+10+12 and 1+5+6+9+14 is 35, that is 1/3 of (0+1+2+3+4+5+6+7+8+9+10+11+12+13+14=) 105.
 
 
There is a more difficult method to produce a 15x15 panmagic square (use the digits 0 up to 14 instead of 1 up to
15, because it is easier to calculate with).
 
The difficulty is the construction of the first row (after that it is not difficult anymore). The key to produce the
first row is the rectangle of 3x5 or 5x3:
 
 
  Rectangle 3x5                                              =           Rectangle 5x3
0
9
12
 
21
 
 
0
9
12
 
21
1
14
6
 
21
 
 
1
14
6
 
21
11
2
8
 
21
 
 
11
2
8
 
21
13
3
5
 
21
 
 
13
3
5
 
21
10
7
4
 
21
 
 
10
7
4
 
21
 
 
 
 
 
 
 
 
 
 
 
 
35
35
35
 
 
 
 
35
35
35
 
 
 
 
The magic sum of 0 up to14 is 105. Notify that in the rectangle the sum of each column of 5 digits is (5/15 x 105 =) 35
and the sum of each row of 3 digits is (3/15 x 105 =) 21. Fill in the digits as follows:
 
 
  Fill in the first row acording to rectangle 3x5
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
 
 
  Fill in the first row according to rectangle 5x3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
 
 
 
Rows 2 up to15 can be produced by moving the first row each time 4 places to the left.
 
 
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
 
 
 
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
 
 
 
 
 
 
 
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
 
 
 
 
 
 
 
 
 
 
 
 
 
 
We have now produced the 1st square with the column coordinates.
 
 
  1st square with column coordinates (take 15x digit + 1)
0
8
7
1
5
9
11
4
14
13
12
2
10
6
3
5
9
11
4
14
13
12
2
10
6
3
0
8
7
1
14
13
12
2
10
6
3
0
8
7
1
5
9
11
4
10
6
3
0
8
7
1
5
9
11
4
14
13
12
2
8
7
1
5
9
11
4
14
13
12
2
10
6
3
0
9
11
4
14
13
12
2
10
6
3
0
8
7
1
5
13
12
2
10
6
3
0
8
7
1
5
9
11
4
14
6
3
0
8
7
1
5
9
11
4
14
13
12
2
10
7
1
5
9
11
4
14
13
12
2
10
6
3
0
8
11
4
14
13
12
2
10
6
3
0
8
7
1
5
9
12
2
10
6
3
0
8
7
1
5
9
11
4
14
13
3
0
8
7
1
5
9
11
4
14
13
12
2
10
6
1
5
9
11
4
14
13
12
2
10
6
3
0
8
7
4
14
13
12
2
10
6
3
0
8
7
1
5
9
11
2
10
6
3
0
8
7
1
5
9
11
4
14
13
12
 
 
The 2nd square is the 1st square, rotated a quarter turn to the left.
 
 
  2nd square with row coordinates (take 1x digit)
3
1
4
2
0
5
14
10
8
9
13
6
7
11
12
6
7
11
12
3
1
4
2
0
5
14
10
8
9
13
10
8
9
13
6
7
11
12
3
1
4
2
0
5
14
2
0
5
14
10
8
9
13
6
7
11
12
3
1
4
12
3
1
4
2
0
5
14
10
8
9
13
6
7
11
13
6
7
11
12
3
1
4
2
0
5
14
10
8
9
14
10
8
9
13
6
7
11
12
3
1
4
2
0
5
4
2
0
5
14
10
8
9
13
6
7
11
12
3
1
11
12
3
1
4
2
0
5
14
10
8
9
13
6
7
9
13
6
7
11
12
3
1
4
2
0
5
14
10
8
5
14
10
8
9
13
6
7
11
12
3
1
4
2
0
1
4
2
0
5
14
10
8
9
13
6
7
11
12
3
7
11
12
3
1
4
2
0
5
14
10
8
9
13
6
8
9
13
6
7
11
12
3
1
4
2
0
5
14
10
0
5
14
10
8
9
13
6
7
11
12
3
1
4
2
 
 
Take a digit from the 1st square multiplied by 15, add 1, and add (1x) the digit from the same cell of the 2nd square
and you get the 15x15 panmagic square as mentioned below.
 
 
 15x15 panmagic square
4
122
110
18
76
141
180
71
219
205
194
37
158
102
58
82
143
177
73
214
197
185
33
151
96
60
11
129
115
29
221
204
190
44
157
98
57
13
124
107
20
78
136
171
75
153
91
51
15
131
114
25
89
142
173
72
223
199
182
35
133
109
17
80
138
166
66
225
206
189
40
164
97
53
12
149
172
68
222
208
184
32
155
93
46
6
135
116
24
85
210
191
39
160
104
52
8
132
118
19
77
140
168
61
216
95
48
1
126
120
26
84
145
179
67
218
207
193
34
152
117
28
79
137
170
63
211
201
195
41
159
100
59
7
128
175
74
217
203
192
43
154
92
50
3
121
111
30
86
144
186
45
161
99
55
14
127
113
27
88
139
167
65
213
196
47
5
123
106
21
90
146
174
70
224
202
188
42
163
94
23
87
148
169
62
215
198
181
36
165
101
54
10
134
112
69
220
209
187
38
162
103
49
2
125
108
16
81
150
176
31
156
105
56
9
130
119
22
83
147
178
64
212
200
183
 
 
The method of construction can be used for each odd multiple of 3, but no multiple of 9 (= 15x15, 21x21, 33x33,
39x39, …). For example to produce a 21x21 pan magic square you can use the following rectangle:
 
 
Rectangle 21x21 panmagic square
0
17
13
 
30
1
9
20
 
30
16
12
2
 
30
19
8
3
 
30
15
11
4