The 3 basic 4x4 panmagic squares
48 of the 880 pure magic 4x4 squares are panmagic (= group 1). These squares have (just like the larger most perfect magic squares) the following structure:
|
1 |
8 |
10 |
15 |
|
12 |
13 |
3 |
6 |
|
7 |
2 |
16 |
9 |
|
14 |
11 |
5 |
4 |
The sum
of two numbers of the same colour is each time (the lowest plus the highest number of the magic square: 1+16=) 17.
You need to know only the following panmagic 4x4 basic squares, to construct all 48 panmagic 4x4 squares (excluding rotation and/or mirroring):
|
1 |
8 |
13 |
12 |
1 |
8 |
11 |
14 |
1 |
8 |
10 |
15 |
||||
|
15 |
10 |
3 |
6 |
15 |
10 |
5 |
4 |
14 |
11 |
5 |
4 |
||||
|
4 |
5 |
16 |
9 |
6 |
3 |
16 |
9 |
7 |
2 |
16 |
9 |
||||
|
14 |
11 |
2 |
7 |
12 |
13 |
2 |
7 |
12 |
13 |
3 |
6 |
Make a 2x2 carpet of one of the basic squares and you can get all 16 (x 3 = 48) squares by shifting on the carpet. See for example:
|
1 |
8 |
10 |
15 |
1 |
8 |
10 |
15 |
|
12 |
13 |
3 |
6 |
12 |
13 |
3 |
6 |
|
7 |
2 |
16 |
9 |
7 |
2 |
16 |
9 |
|
14 |
11 |
5 |
4 |
14 |
11 |
5 |
4 |
|
1 |
8 |
10 |
15 |
1 |
8 |
10 |
15 |
|
12 |
13 |
3 |
6 |
12 |
13 |
3 |
6 |
|
7 |
2 |
16 |
9 |
7 |
2 |
16 |
9 |
|
14 |
11 |
5 |
4 |
14 |
11 |
5 |
4 |
You can get 8 results in stead of 1 result of the above yellow marked square by rotating and/or mirroring:
|
yellow marked square |
4 |
14 |
11 |
5 |
Mirroring |
5 |
11 |
14 |
4 |
||||||
|
15 |
1 |
8 |
10 |
10 |
8 |
1 |
15 |
||||||||
|
6 |
12 |
13 |
3 |
3 |
13 |
12 |
6 |
||||||||
|
9 |
7 |
2 |
16 |
16 |
2 |
7 |
9 |
||||||||
|
rotation by 1 quarter |
9 |
6 |
15 |
4 |
Mirroring |
4 |
15 |
6 |
9 |
||||||
|
7 |
12 |
1 |
14 |
14 |
1 |
12 |
7 |
||||||||
|
2 |
13 |
8 |
11 |
11 |
8 |
13 |
2 |
||||||||
|
16 |
3 |
10 |
5 |
5 |
10 |
3 |
16 |
||||||||
|
rotation by 2 quarters |
16 |
2 |
7 |
9 |
Mirroring |
9 |
7 |
2 |
16 |
||||||
|
3 |
13 |
12 |
6 |
6 |
12 |
13 |
3 |
||||||||
|
10 |
8 |
1 |
15 |
15 |
1 |
8 |
10 |
||||||||
|
5 |
11 |
14 |
4 |
4 |
14 |
11 |
5 |
||||||||
|
rotation by 3 quarters |
5 |
10 |
3 |
16 |
Mirroring |
16 |
3 |
10 |
5 |
||||||
|
11 |
8 |
13 |
2 |
2 |
13 |
8 |
11 |
||||||||
|
14 |
1 |
12 |
7 |
7 |
12 |
1 |
14 |
||||||||
|
4 |
15 |
6 |
9 |
9 |
6 |
15 |
4 |
||||||||
There are 3 basic 4x4 panmagic squares. There are 16 possibilities by shifting on the carpet. There are 8 possibilities by rotating and/or mirroring. This gives in total 3 x 16 x 8 is 384 possibilities (including rotating and/or mirroring).
