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How to make perfect magic squares, (hyper)cubes, etcetera
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Sudoku method (3)
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How to use only one 4x4 Sudoku to produce (in 9 steps) a most perfect
magic 1024x1024 square
 

 
  4x4 Sudoku
2
1
3
0
1
2
0
3
0
3
1
2
3
0
2
1
 

 
Step 1
 
First we use the 4x4 Sudoku to produce a 4x4 panmagic square. You need the 4x4 Sudoku plus an - on the
2x2 carpet shifted (1 tot the right and 1 down) - version of the same 4x4 Sudoku.
 
 
 4x4 Sudoku shifted on the 2x2 carpet
2
1
3
0
2
1
3
0
1
2
0
3
1
2
0
3
0
3
1
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3
1
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3
0
2
1
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2
1
2
1
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2
1
3
0
1
2
0
3
1
2
0
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3
1
2
0
3
1
2
3
0
2
1
3
0
2
1
 
 
Take 4x a digit from the 4x4 Sudoku and add 1x a digit of the same cell from the shifted 4x4 Sudoku and add 1
to each digit to produce a 4x4 panmagic square.
 
 
  4x digit                 +    1x digit                =               +1             = 4x4 panmagic square
2
1
3
0
 
2
0
3
1
 
10
4
15
1
 
11
5
16
2
1
2
0
3
 
3
1
2
0
 
7
9
2
12
 
8
10
3
13
0
3
1
2
 
0
2
1
3
 
0
14
5
11
 
1
15
6
12
3
0
2
1
 
1
3
0
2
 
13
3
8
6
 
14
4
9
7

 
 
Step 2
 
Use the 4x4 panmagic square to produce a 8x8 most perfect (Franklin pan)magic square. You need the 2x2 car-
pet of the 4x4 panmagic square plus an 8x8 Sudoku pattern.
 
Use the 4x4 Sudoku to produce the 8x8 Sudoku pattern. Put next to the 4x4 Sudoku a second 4x4 Sudoku by switching
the left and the right half. Put below the first and the second 4x4 Sudoku a third and a fourth 4x4 Sudoku by switching
the top and the bottom half.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Take 1x a digit from the 2x2 carpet of the 4x4 panmagic square and add 16x a digit of the same cell from the 8x8 Sudoku
pattern to produce a 8x8 most perfect (Franklin pan)magic square.
 
 
  1x digit                                         +      16x digit                             =  8x8 most perfect (Franklin pan)magic
11
5
16
2
11
5
16
2
 
 
2
1
3
0
3
0
2
1
 
 
43
21
64
2
59
5
48
18
8
10
3
13
8
10
3
13
 
 
1
2
0
3
0
3
1
2
 
 
24
42
3
61
8
58
19
45
1
15
6
12
1
15
6
12
 
 
0
3
1
2
1
2
0
3
 
 
1
63
22
44
17
47
6
60
14
4
9
7
14
4
9
7
 
 
3
0
2
1
2
1
3
0
 
 
62
4
41
23
46
20
57
7
11
5
16
2
11
5
16
2
 
 
0
3
1
2
1
2
0
3
 
 
11
53
32
34
27
37
16
50
8
10
3
13
8
10
3
13
 
 
3
0
2
1
2
1
3
0
 
 
56
10
35
29
40
26
51
13
1
15
6
12
1
15
6
12
 
 
2
1
3
0
3
0
2
1
 
 
33
31
54
12
49
15
38
28
14
4
9
7
14
4
9
7
 
 
1
2
0
3
0
3
1
2
 
 
30
36
9
55
14
52
25
39
 
 
 
Step 3
 
Use the 8x8 most perfect (Franklin pan)magic square to produce a most perfect 16x16 (Franklin pan)magic square.
You need the 2x2 carpet of the 8x8 most perfect (Franklin pan)magic square plus an 16x16 Sudoku pattern.
 
Use the 8x8 Sudoku pattern to produce the 16x16 Sudoku pattern. Put next to the 8x8 Sudoku pattern a second 8x8
Sudoku pattern by switching the left and the right half. Put below the first and the second 8x8 Sudoku pattern a third
and a fourth 8x8 Sudoku pattern by switching the top and the bottom half.
 
Take 1x a digit from the 2x2 carpet of the 8x8 Franklin panmagic square and add (16 x 4 = ) 64x a digit of the same
cell from the 16x16 Sudoku pattern to produce a most perfect 16x16 (Franklin pan)magic square.
 
 
  1x digit
43
21
64
2
59
5
48
18
43
21
64
2
59
5
48
18
24
42
3
61
8
58
19
45
24
42
3
61
8
58
19
45
1
63
22
44
17
47
6
60
1
63
22
44
17
47
6
60
62
4
41
23
46
20
57
7
62
4
41
23
46
20
57
7
11
53
32
34
27
37
16
50
11
53
32
34
27
37
16
50
56
10
35
29
40
26
51
13
56
10
35
29
40
26
51
13
33
31
54
12
49
15
38
28
33
31
54
12
49
15
38
28
30
36
9
55
14
52
25
39
30
36
9
55
14
52
25
39
43
21
64
2
59
5
48
18
43
21
64
2
59
5
48
18
24
42
3
61
8
58
19
45
24
42
3
61
8
58
19
45
1
63
22
44
17
47
6
60
1
63
22
44
17
47
6
60
62
4
41
23
46
20
57
7
62
4
41
23
46
20
57
7
11
53
32
34
27
37
16
50
11
53
32
34
27
37
16
50
56
10
35
29
40
26
51
13
56
10
35
29
40
26
51
13
33
31
54
12
49
15
38
28
33
31
54
12
49
15
38
28
30
36
9
55
14
52
25
39
30
36
9
55
14
52
25
39
 
 
+
 
 
  64x digit
2
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3
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3
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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
 
 
=
 
 
  Most perfect 16x16 (Franklin pan)magic square
171
85
256
2
251
5
176
82
235
21
192
66
187
69
240
18
88
170
3
253
8
250
83
173
24
234
67
189
72
186
19
237
1
255
86
172
81
175
6
252
65
191
22
236
17
239
70
188
254
4
169
87
174
84
249
7
190
68
233
23
238
20
185
71
11
245
96
162
91
165
16
242
75
181
32
226
27
229
80
178
248
10
163
93
168
90
243
13
184
74
227
29
232
26
179
77
161
95
246
12
241
15
166
92
225
31
182
76
177
79
230
28
94
164
9
247
14
244
89
167
30
228
73
183
78
180
25
231
43
213
128
130
123
133
48
210
107
149
64
194
59
197
112
146
216
42
131
125
136
122
211
45
152
106
195
61
200
58
147
109
129
127
214
44
209
47
134
124
193
63
150
108
145
111
198
60
126
132
41
215
46
212
121
135
62
196
105
151
110
148
57
199
139
117
224
34
219
37
144
114
203
53
160
98
155
101
208
50
120
138
35
221
40
218
115
141
56
202
99
157
104
154
51
205
33
223
118
140
113
143
38
220
97
159
54
204
49
207
102
156
222
36
137
119
142
116
217
39
158
100
201
55
206
52
153
103
 

 
Step 4 up to 9
 
Repeat step 3, six times . Produce successively a most perfect (Franklin pan)magic 32x32, 64x64, 128x128, 256x256,
512x512 and 1024x1024 square.
 
 
Notify that all used squares and Sudoku patterns are products from the same 4x4 Sudoku.

The 4x4 Sudoku is a ‘duplicater’. I found the following 32 duplicaters:
 
 

1
 
0
3
1
2
 
 
2
 
3
1
2
0
 
 
3
 
1
2
0
3
 
 
4
 
2
0
3
1
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
5
 
3
0
2
1
 
 
6
 
0
2
1
3
 
 
7
 
2
1
3
0
 
 
8
 
1
3
0
2
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
9
 
2
1
3
0
 
 
10
 
1
3
0
2
 
 
11
 
3
0
2
1
 
 
12
 
0
2
1
3
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
13
 
1
2
0
3
 
 
14
 
2
0
3
1
 
 
15
 
0
3
1
2
 
 
16
 
3
1
2
0
 
 
0
3
1
2
 
 
 
 
3
1
2
0
 
 
 
 
1
2
0
3
 
 
 
 
2
0
3
1
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
2
1
3
0
 
 
 
 
1
3
0
2
 
 
 
 
3
0
2
1
 
 
 
 
0
2
1
3
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
17
 
2
1
0
3
 
 
18
 
1
0
3
2
 
 
19
 
0
3
2
1
 
 
20
 
3
2
1
0
 
 
0
3
2
1
 
 
 
 
3
2
1
0
 
 
 
 
2
1
0
3
 
 
 
 
1
0
3
2
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
 
 
1
2
3
0
 
 
 
 
2
3
0
1
 
 
1
2
3
0
 
 
 
 
2
3
0
1
 
 
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
21
 
0
3
2
1
 
 
22
 
3
2
1
0
 
 
23
 
2
1
0
3
 
 
24
 
1
0
3
2
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
 
 
1
2
3
0
 
 
 
 
2
3
0
1
 
 
1
2
3
0
 
 
 
 
2
3
0
1
 
 
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
2
1
0
3
 
 
 
 
1
0
3
2
 
 
 
 
0
3
2
1
 
 
 
 
3
2
1
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
25
 
3
0
1
2
 
 
26
 
0
1
2
3
 
 
27
 
1
2
3
0
 
 
28
 
2
3
0
1
 
 
1
2
3
0
 
 
 
 
2
3
0
1
 
 
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
2
1
0
3
 
 
 
 
1
0
3
2
 
 
 
 
0
3
2
1
 
 
 
 
3
2
1
0
 
 
0
3
2
1
 
 
 
 
3
2
1
0
 
 
 
 
2
1
0
3
 
 
 
 
1
0
3
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
29