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Perfect magic squares
Pan magic 27x27 square
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How to produce a pan magic 27x27 square
 
 
A 27x27 square is a multiple of 3 (and the size is no odd number squared), but still panmagic 27x27 squares
exist. See below a method of construction (do not fill in the digits 1 up to 27, but fill in the digits 0 up to 26,
because it is easier to calculate with).
 
Produce first 1/3 of the first row (27 / 3 = 9 digits). Take for example the digits 0 up to 8 in the same sequence.
Then produce 1/3 of the second row with the digits 9 up to 17 and 1/3 of the third row with the digits 18 up to
26. The sum of each (1/9) column must be ([0 t/m 26] / 9 = ) 39.
 
 
  1/3 of 1st/2nd/3rd row (sum 1/9 column is 39)
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
39
39
39
39
39
39
39
39
39
 
 
Then finish the first three rows by using the same 1/3 rows in a different sequence of (top down ) 2-3-1 and
3-1-2.
 
 
  First three rows of the 1st square
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
 
 
Copy the first three rows to the bottom until the square has been filled completely. The 1st square consist of
row coordinates.
 
 
  1st square with the row coordinates (take a digit [x 1])
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
17
15
16
10
11
9
14
12
13
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
22
23
21
26
24
25
19
20
18
0
1
2
3
4
5
6
7
8
17
15
16
10
11
9
14
12
13
 
 
Produce the 2nd square with the column coordinates by rotating the 1st square by a quarter turn to the right and
by mirroring (verticaly).
 
 
  2nd square with the column coordinates (take a digit x 27 + 1)
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
6
14
19
6
14
19
6
14
19
6
14
19
6
14
19
6
14
19
6
14
19
6
14
19
6
14
19
7
12
20
7
12
20
7
12
20
7
12
20
7
12
20
7
12
20
7
12
20
7
12
20
7
12
20
8
13
18
8
13
18
8
13
18
8
13
18
8
13
18
8
13
18
8
13
18
8
13
18
8
13
18
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
17
22
0
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
15
23
1
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
16
21
2
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
10
26
3
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
11
24
4
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
9
25
5
14
19
6
14