Magic squares and cubesContact / guestbookMost magic square per order3x3 magic square3x3 magic square, explanationSudoku method (1)Sudoku method (2)Sudoku method (3)Pan magic 4x4 squarePan magic 4x4 square, explanationPan magic 4x4 square, binaryDürer & Franklin transformationTransformation methodTransformation method, analysis[ultra] pan magic 5x5 squarePan magic 5x5 square, explanation6x6 magic squareUltra (pan)magic 8x8 squareMost perfect magic squares, explanation8x8 most perfect magic squares, binaryKhajuraho methodKhajuraho method, explanationBasic pattern method (1a)Basic pattern method (1b)Basic pattern method (2)Basic pattern method (3a)Basic pattern method (3b)Basic pattern method (3c)Basic pattern method (4)Basic pattern method (5)Basic pattern method (6)Basic pattern method (7a)Basic pattern method (7b)Analysis Franklin panm. 8x8 (1)Analysis Franklin panm. 8x8 (2)Basic key method (1)Basic key method (2)Quadrant method (Willem Barink)Quadrant method group 1 up to 5Quadrant method group 6 up to 10Quadrant method group 11 up to 19[ultra] pan magic 9x9 square (1)pan magic 9x9 square (2)pan magic 9x9 square (3)3x extra magic 9x9 squarebimagic 9x9 square10x10 magic squareComposite 12x12 magic square14x14 magic square[Ultra] pan magic 15x15 square3x extra magic 15x15 squareThe perfect magic square3x extra magic 18x18 squareComposite 24x24 magic squareUltra pan magic 25x25 square (1)Ultra pan magic 25x25 square (2)Ultra bimagic 25x25 square[ultra] pan magic 27x27 square[ultra] pan magic 35x35 squareextra magic 35x35 squareBordered squaresInlaid square (1)Inlaid square (2)Each magic sumWater retention challengeMost magic cube per orderMost magic 4x4x4 cubesymmetric & semi (pan)magic 5x5x5 cubeSymmetric & panmagic 7x7x7 cubePerfect (Nasik) & compact 8x8x8 cube[More than] perfect magic 9x9x9 cubePerfect (Nasik) magic 11x11x11 cubePerfect (Nasik) magic 15x15x15 cubeTrick with 8x8 bimagic squarePerfect (Nasik) 16x16x16, step 1&2Perfect (Nasik) 16x16x16, step 3&4Perfect (Nasik) 16x16x16, resultFavorite Links
How to make perfect magic squares & cubes
The sky is the limit!!!
Sudoku method (3)
<HOME>                         <<PREVIOUS]                         [NEXT>>



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
2
0
3
1
2
3
0
2
1
3
0
2
1
2
1
3
0
2
1
3
0
1
2
0
3
1
2
0
3
0
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
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
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
 
1
2
3
0
 
 
30
 
2
3
0
1
 
 
31
 
3
0
1
2
 
 
32
 
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
 
 
3
0
1
2
 
 
 
 
0
1
2
3
 
 
 
 
1
2
3
0
 
 
 
 
2
3
0
1

 
 
There are 384 panmagic 4x4 squares. So with the 32 duplicaters there are 384 x 32 is 12288 possibilities to produce a most
perfect (Franklin pan)magic 8x8 square.
 
It is also possible to swap row 1&3 and/or row 2&4 and/or row 5&7 and/or row 6&8 and/or column 1&3 and/or column 2&4
and/or column 5&7 and/or column 6&8 (that gives 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 options). Swapping rows and/or columns
gives a lot (but unknown number) of double solutions.


For analysis and construction the following downloads in EXCEL format are available:


See for a complete classification of all most perfect (Franklin pan)magic 8x8 squares:  'Analysis Franklin panmagic 8x8 (2)'




<HOME>                         <<PREVIOUS]                         [NEXT>>

Magic squares and cubes|Contact / guestbook|Most magic square per order|3x3 magic square|3x3 magic square, explanation|Sudoku method (1)|Sudoku method (2)|Sudoku method (3)|Pan magic 4x4 square|Pan magic 4x4 square, explanation|Pan magic 4x4 square, binary|Dürer & Franklin transformation|Transformation method|Transformation method, analysis|[ultra] pan magic 5x5 square|Pan magic 5x5 square, explanation|6x6 magic square|Ultra (pan)magic 8x8 square|Most perfect magic squares, explanation|8x8 most perfect magic squares, binary|Khajuraho method|Khajuraho method, explanation|Basic pattern method (1a)|Basic pattern method (1b)|Basic pattern method (2)|Basic pattern method (3a)|Basic pattern method (3b)|Basic pattern method (3c)|Basic pattern method (4)|Basic pattern method (5)|Basic pattern method (6)|Basic pattern method (7a)|Basic pattern method (7b)|Analysis Franklin panm. 8x8 (1)|Analysis Franklin panm. 8x8 (2)|Basic key method (1)|Basic key method (2)|Quadrant method (Willem Barink)|Quadrant method group 1 up to 5|Quadrant method group 6 up to 10|Quadrant method group 11 up to 19|[ultra] pan magic 9x9 square (1)|pan magic 9x9 square (2)|pan magic 9x9 square (3)|3x extra magic 9x9 square|bimagic 9x9 square|10x10 magic square|Composite 12x12 magic square|14x14 magic square|[Ultra] pan magic 15x15 square|3x extra magic 15x15 square|The perfect magic square|3x extra magic 18x18 square|Composite 24x24 magic square|Ultra pan magic 25x25 square (1)|Ultra pan magic 25x25 square (2)|Ultra bimagic 25x25 square|[ultra] pan magic 27x27 square|[ultra] pan magic 35x35 square|extra magic 35x35 square|Bordered squares|Inlaid square (1)|Inlaid square (2)|Each magic sum|Water retention challenge|Most magic cube per order|Most magic 4x4x4 cube|symmetric & semi (pan)magic 5x5x5 cube|Symmetric & panmagic 7x7x7 cube|Perfect (Nasik) & compact 8x8x8 cube|[More than] perfect magic 9x9x9 cube|Perfect (Nasik) magic 11x11x11 cube|Perfect (Nasik) magic 15x15x15 cube|Trick with 8x8 bimagic square|Perfect (Nasik) 16x16x16, step 1&2|Perfect (Nasik) 16x16x16, step 3&4|Perfect (Nasik) 16x16x16, result|Favorite Links