Symmetric transformation
Marios Mamzeris shows us that you can transform a odd square with sequencial numbers into a symmetric magic square in two steps (https://www.youtube.com/watch?v=JPEB8aT7XiU):
Step 1, horizontal swap
| 1 | 2 | 3 | > | 3 | 1 | 2 | ||
| 4 | 5 | 6 | 4 | 5 | 6 | |||
| < | 7 | 8 | 9 | 8 | 9 | 7 |
Step 2, vertical swap
| ^ | ||||||||
| 3 | 1 | 2 | 8 | 1 | 6 | |||
| 4 | 5 | 6 | 3 | 5 | 7 | |||
| 8 | 9 | 7 | 4 | 9 | 2 | |||
| v |
You can use this method to construct magic squares of odd order from 3x3 to infinite and you get a symmetric (but not pan)magic square.
See 3x3, 5x5, 7x7, 9x9, 11x11, 13x13, 15x15, 17x17, 19x19, 21x21, 23x23, 25x25, 27x27, 29x29 and 31x31
