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How to make perfect magic squares & cubes
The sky is the limit!!!
3x extra magic 18x18 square
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Method 1,  9x sequencial 6x6 and structure of 3x3 to produce 18x18
magic square

Take 1x a digit from the pattern with 3x3 the same magic 6x6 square and add 36x a [digit minus 1]
from the same cell of the (6x6 ‘blown up´) pattern of a magic 3x3 square.
 
 
+1x digit from 3x3 the same magic 6x6 square
 
 
 
 
 
 
20
29
9
27
4
22
20
29
9
27
4
22
20
29
9
27
4
22
11
2
36
18
31
13
11
2
36
18
31
13
11
2
36
18
31
13
34
16
14
23
21
3
34
16
14
23
21
3
34
16
14
23
21
3
7
25
5
32
30
12
7
25
5
32
30
12
7
25
5
32
30
12
33
24
19
1
8
26
33
24
19
1
8
26
33
24
19
1
8
26
6
15
28
10
17
35
6
15
28
10
17
35
6
15
28
10
17
35
20
29
9
27
4
22
20
29
9
27
4
22
20
29
9
27
4
22
11
2
36
18
31
13
11
2
36
18
31
13
11
2
36
18
31
13
34
16
14
23
21
3
34
16
14
23
21
3
34
16
14
23
21
3
7
25
5
32
30
12
7
25
5
32
30
12
7
25
5
32
30
12
33
24
19
1
8
26
33
24
19
1
8
26
33
24
19
1
8
26
6
15
28
10
17
35
6
15
28
10
17
35
6
15
28
10
17
35
20
29
9
27
4
22
20
29
9
27
4
22
20
29
9
27
4
22
11
2
36
18
31
13
11
2
36
18
31
13
11
2
36
18
31
13
34
16
14
23
21
3
34
16
14
23
21
3
34
16
14
23
21
3
7
25
5
32
30
12
7
25
5
32
30
12
7
25
5
32
30
12
33
24
19
1
8
26
33
24
19
1
8
26
33
24
19
1
8
26
6
15
28
10
17
35
6
15
28
10
17
35
6
15
28
10
17
35
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+36x [digit minus 1] from the (6x6 'blown up') magic 3x3 square
 
 
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
8
8
8
8
8
8
1
1
1
1
1
1
6
6
6
6
6
6
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
3
3
3
3
3
3
5
5
5
5
5
5
7
7
7
7
7
7
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
4
4
4
4
4
4
9
9
9
9
9
9
2
2
2
2
2
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
= magic 18x18 square (consisting of 9 magic 6x6 squares)
 
 
272
281
261
279
256
274
20
29
9
27
4
22
200
209
189
207
184
202
263
254
288
270
283
265
11
2
36
18
31
13
191
182
216
198
211
193
286
268
266
275
273
255
34
16
14
23
21
3
214
196
194
203
201
183
259
277
257
284
282
264
7
25
5
32
30
12
187
205
185
212
210
192
285
276
271
253
260
278
33
24
19
1
8
26
213
204
199
181
188
206
258
267
280
262
269
287
6
15
28
10
17
35
186
195
208
190
197
215
92
101
81
99
76
94
164
173
153
171
148
166
236
245
225
243
220
238
83
74
108
90
103
85
155
146
180
162
175
157
227
218
252
234
247
229
106
88
86
95
93
75
178
160
158
167
165
147
250
232
230
239
237
219
79
97
77
104
102
84
151
169
149
176
174
156
223
241
221
248
246
228
105
96
91
73
80
98
177
168
163
145
152
170
249
240
235
217
224
242
78
87
100
82
89
107
150
159
172
154
161
179
222
231
244
226
233
251
128
137
117
135
112
130
308
317
297
315
292
310
56
65
45
63
40
58
119
110
144
126
139
121
299
290
324
306
319
301
47
38
72
54
67
49
142
124
122
131
129
111
322
304
302
311
309
291
70
52
50
59
57
39
115
133
113
140
138
120
295
313
293
320
318
300
43
61
41
68
66
48
141
132
127
109
116
134
321
312
307
289
296
314
69
60
55
37
44
62
114
123
136
118
125
143
294
303
316
298
305
323
42
51
64
46
53
71
 
 
 
Method 2, classic 16x16 inlay with border to produce 18x18 magic square

First we make the 16x16 (inlay) square by combining digits from two patterns with the digits 1 up to 256
and (counting down from) 256 up to 1.

 
16x16 square with the digits 1 up to 256
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
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
16x16 square with the digits (counting down from) 256 up to 1
256
255
254
253
252
251
250
249
248
247
246
245
244
243
242
241
240
239
238
237
236
235
234
233
232
231
230
229
228
227
226
225
224
223
222
221
220
219
218
217
216
215
214
213
212
211
210
209
208
207
206
205
204
203
202
201
200
199
198
197
196
195
194
193
192
191
190
189
188
187
186
185
184
183
182
181
180
179
178
177
176
175
174
173
172
171
170
169
168
167
166
165
164
163
162
161
160
159
158
157
156
155
154
153
152
151
150
149
148
147
146
145
144
143
142
141
140
139
138
137
136
135
134
133
132
131
130
129
128
127
126
125
124
123
122
121
120
119
118
117
116
115
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
= magic 16x16 square (as combination of the squares above)
1
255
254
4
5
251
250
8
9
247
246
12
13
243
242
16
240
18
19
237
236
22
23
233
232
26
27
229
228
30
31
225
224
34
35
221
220
38
39
217
216
42
43
213
212
46
47
209
49
207
206
52
53
203
202
56
57
199
198
60
61
195
194
64
65
191
190
68
69
187
186
72
73
183
182
76
77
179
178
80
176
82
83
173
172
86
87
169
168
90
91
165
164
94
95
161
160
98
99
157
156
102
103
153
152
106
107
149
148
110
111
145
113
143
142
116
117
139
138
120
121
135
134
124
125
131
130
128
129
127
126
132
133
123
122
136
137
119
118
140
141
115
114
144
112
146
147
109
108
150
151
105
104
154
155
101
100
158
159
97
96
162
163
93
92
166
167
89
88
170
171
85
84
174
175
81
177
79
78
180
181
75
74
184
185
71
70
188
189
67
66
192
193
63
62
196
197
59
58
200
201
55
54
204
205
51
50
208
48
210
211
45
44
214
215
41
40
218
219
37
36
222
223
33
32
226
227
29
28
230
231
25
24
234
235
21
20
238
239
17
241
15
14
244
245
11
10
248
249
7
6
252
253
3
2
256
 
 
Produce a concentric 18x18 magic square on website http://users.eastlink.ca/~sharrywhite/Bordered
MagicSquares.html, and use only the (grey marked) border. Because the border consists of the
digits 1 up to 34 (and 291 up to 324), you need to add 34 to all digits of the 16x16 (inlay) square.
 
 
16x16 in 18x18 magic square
 307
25
301
23
303
21
305
19
316
1
309
15
311
13
313
11
315
17
298
35
289
288
38
39
285
284
42
43
281
280
46
47
277
276
50
27
28
274
52
53
271
270
56
57
267
266
60
61
263
262
64
65
259
297
296
258
68
69
255
254
72
73
251
250
76
77
247
246
80
81
243
29
30
83
241
240
86
87
237
236
90
91
233
232
94
95
229
228
98
295
294
99
225
224
102
103
221
220
106
107
217
216
110
111
213
212
114
31
32
210
116
117
207
206
120
121
203
202
124
125
199
198
128
129
195
293
292
194
132
133
191
190
136
137
187
186
140
141
183
182
144
145
179
33
34
147
177
176
150
151
173
172
154
155
169
168
158
159
165
164
162
291
26
163
161
160
166
167
157
156
170
171
153
152
174
175
149
148
178
299
8
146
180
181
143
142
184
185
139
138
188
189
135
134
192
193
131
317
318
130
196
197
127
126
200
201
123
122
204
205
119
118
208
209
115
7
6
211
113
112
214
215
109
108
218
219
105
104
222
223
101
100
226
319
320
227
97
96
230
231
93
92
234
235
89
88
238
239
85
84
242
5
4
82
244
245
79
78
248
249
75
74
252
253
71
70
256
257
67
321
322
66
260
261
63
62
264
265
59
58
268
269
55
54
272
273
51
3
2
275
49
48
278
279
45
44
282
283
41
40
286
287
37
36
290
323
308
300
24
302
22
304
20
306
9
324
16
310
14
312
12
314
10
 18
 
 
 
Method 3, 9x magic 4x4 in 6x6 square to produce 18x18 magic square 

The first grid is 9x the same magic 4x4 in 6x6 square. The second grid is fixed. Take 1x a digit from the
first grid and add 36x a digit from the same cell of the second grid.
 

1x digit from grid with 9x the same 4x4 in 6x6
 
 
 
 
 
 
1
6
9
34
32
29
1
6
9
34
32
29
1
6
9
34
32
29
35
11
18
23
22
2
35
11
18
23
22
2
35
11
18
23
22
2
33
25
20
13
16
4
33
25
20
13
16
4
33
25
20
13
16
4
27
14
15
26
19
10
27
14
15
26
19
10
27
14
15
26
19
10
7
24
21
12
17
30
7
24
21
12
17
30
7
24
21
12
17
30
8
31
28
3
5
36
8
31
28
3
5
36
8
31
28
3
5
36
1
6
9
34
32
29
1
6
9
34
32
29
1
6
9
34
32
29
35
11
18
23
22
2
35
11
18
23
22
2
35
11
18
23
22
2
33
25
20
13
16
4
33
25
20
13
16
4
33
25
20
13
16
4
27
14
15
26
19
10
27
14
15
26
19
10
27
14
15
26
19
10
7
24
21
12
17
30
7
24
21
12
17
30
7
24
21
12
17
30
8
31
28
3
5
36
8
31
28
3
5
36
8
31
28
3
5
36
1
6
9
34
32
29
1
6
9
34
32
29
1
6
9
34
32
29
35
11
18
23
22
2
35
11
18
23
22
2
35
11
18
23
22
2
33
25
20
13
16
4
33
25
20
13
16
4
33
25
20
13
16
4
27
14
15
26
19
10
27
14
15
26
19
10
27
14
15
26
19
10
7
24
21
12
17
30
7
24
21
12
17
30
7
24
21
12
17
30
8
31
28
3
5
36
8
31
28
3
5
36
8
31
28
3
5
36
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 36x digit
 
 
 
 
 
 
 
 
 
 
 
 
8
0
8
0
8
0
7
1
7
1
7
1
6
2
6
2
6
2
0
8
0
8
0
8
1
7
1
7
1
7
2
6
2
6
2
6
8
0
8
0
8
0
7
1
7
1
7
1
6
2
6
2
6
2
0
0
8
0
8
8
1
1
7
1
7
7
2
2
6
2
6
6
0
8
0
8
0
8
1
7
1
7
1
7
2
6
2
6
2
6
8
8
0
8
0
0
7
7
1
7
1
1
6
6
2
6
2
2
5
3
5
3
5
3
4
4
4
4
4
4
3
5
3
5
3
5
3
5
3
5
3
5
4
4
4
4
4
4
5
3
5
3
5
3
5
3
5
3
5
3
4
4
4
4
4
4
3
5
3
5
3
5
3
3
5
3
5
5
4
4
4
4
4
4
5
5
3
5
3
3
3
5
3
5
3
5
4
4
4
4
4
4
5
3
5
3
5
3
5
5
3
5
3
3
4
4
4
4
4
4
3
3
5
3
5
5
2
6
2
6
2
6
1
7
1
7
1
7
0
8
0
8
0
8
6
2
6
2
6
2
7
1
7
1
7
1
8
0
8
0
8
0
2
6
2
6
2
6
1
7
1
7
1
7
0
8
0
8
0
8
6
6
2
6
2
2
7
7
1
7
1
1
8
8
0
8
0
0
6
2
6
2
6
2
7
1
7
1
7
1
8
0
8
0
8
0
2
2
6
2
6
6
1
1
7
1
7
7
0
0
8
0
8
8
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
= 18x18 magic square with 9x proportional 4x4 in 6x6
 
 
 
 
289
6
297
34
320
29
253
42
261
70
284
65
217
78
225
106
248
101
35
299
18
311
22
290
71
263
54
275
58
254
107
227
90
239
94
218
321
25
308
13
304
4
285
61
272
49
268
40
249
97
236
85
232
76
27
14
303
26
307
298
63
50
267
62
271
262
99
86
231
98
235
226
7
312
21
300
17
318
43
276
57
264
53
282
79
240
93
228
89
246
296
319
28
291
5
36
260
283
64
255
41
72
224
247
100
219
77
108
181
114
189
142
212
137
145
150
153
178
176
173
109
186
117
214
140
209
143
191
126
203
130
182
179
155
162
167
166
146
215
119
198
131
202
110
213
133
200
121
196
112
177
169
164
157
160
148
141
205
128
193
124
184
135
122
195
134
199
190
171
158
159
170
163
154
207
194
123
206
127
118
115
204
129
192
125
210
151
168
165
156
161
174
187
132
201
120
197
138
188
211
136
183
113
144
152
175
172
147
149
180
116
139
208
111
185
216
73
222
81
250
104
245
37
258
45
286
68
281
1
294
9
322
32
317
251
83
234
95
238
74
287
47
270
59
274
38
323
11
306
23
310
2
105
241
92
229
88
220
69
277
56
265
52
256
33
313
20
301
16
292
243
230
87
242
91
82
279
266
51
278
55
46
315
302
15
314
19
10
223
96
237
84
233
102
259
60
273
48
269
66
295
24
309
12
305
30
80
103
244
75
221
252
44
67
280
39
257
288
8
31
316
3
293
324
 
 
N.B.: Establish that each 1/3 row, each 1/3 column and each 1/3 diagonal gives 1/3 of the magic sum.

 
You can use this method to produce multiples of 6 (= 12x12, 18x18, 24x24, 30x30, …). To produce an extra
magic 30x30 square: use 25x the same 4x4 in 6x6 magic square.


The following downloads for analysis and construction in EXCEL are available:

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