Ultra magic 31x31 square
The key to construct an ultra magic 31x31 square is 0 - 30 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20 - 21- 22 - 23 - 24 - 25 - 26 - 27 - 28 - 29
The second grid is a reflection (rotated by a quarter and mirrored) of the first grid.
Take 1x number from the first grid +1
| 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
| 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |
| 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
| 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 |
| 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 |
| 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |
| 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |
| 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
| 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 | 5 | 6 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 | 3 | 4 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 | 1 | 2 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 0 | 30 |
+ 31x number from second grid
| 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 |
| 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 |
| 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 |
| 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 |
| 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 |
| 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 |
| 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 |
| 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 |
| 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 |
| 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 |
| 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 |
| 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 |
| 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 |
| 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 |
| 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 |
| 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 |
| 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 |
| 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 |
| 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 |
| 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 |
| 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 |
| 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 |
| 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 |
| 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 |
| 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 |
| 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 |
| 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 |
| 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 |
| 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 |
| 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 | 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 |
| 29 | 27 | 25 | 23 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 0 | 28 | 26 | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 30 |
= Ultra magic 31x31 square
| 1 | 899 | 808 | 747 | 686 | 625 | 564 | 503 | 442 | 381 | 320 | 259 | 198 | 137 | 76 | 945 | 915 | 854 | 793 | 732 | 671 | 610 | 549 | 488 | 427 | 366 | 305 | 244 | 183 | 122 | 61 |
| 959 | 929 | 838 | 806 | 715 | 654 | 593 | 532 | 471 | 410 | 349 | 288 | 227 | 166 | 105 | 44 | 14 | 883 | 822 | 761 | 700 | 639 | 578 | 517 | 456 | 395 | 334 | 273 | 212 | 151 | 90 |
| 58 | 28 | 897 | 836 | 745 | 713 | 622 | 561 | 500 | 439 | 378 | 317 | 256 | 195 | 134 | 73 | 942 | 912 | 851 | 790 | 729 | 668 | 607 | 546 | 485 | 424 | 363 | 302 | 241 | 180 | 119 |
| 87 | 956 | 926 | 865 | 804 | 743 | 652 | 620 | 529 | 468 | 407 | 346 | 285 | 224 | 163 | 102 | 41 | 11 | 880 | 819 | 758 | 697 | 636 | 575 | 514 | 453 | 392 | 331 | 270 | 209 | 148 |
| 116 | 55 | 25 | 894 | 833 | 772 | 711 | 650 | 559 | 527 | 436 | 375 | 314 | 253 | 192 | 131 | 70 | 939 | 909 | 848 | 787 | 726 | 665 | 604 | 543 | 482 | 421 | 360 | 299 | 238 | 177 |
| 145 | 84 | 953 | 923 | 862 | 801 | 740 | 679 | 618 | 557 | 466 | 434 | 343 | 282 | 221 | 160 | 99 | 38 | 8 | 877 | 816 | 755 | 694 | 633 | 572 | 511 | 450 | 389 | 328 | 267 | 206 |
| 174 | 113 | 52 | 22 | 891 | 830 | 769 | 708 | 647 | 586 | 525 | 464 | 373 | 341 | 250 | 189 | 128 | 67 | 936 | 906 | 845 | 784 | 723 | 662 | 601 | 540 | 479 | 418 | 357 | 296 | 235 |
| 203 | 142 | 81 | 950 | 920 | 859 | 798 | 737 | 676 | 615 | 554 | 493 | 432 | 371 | 280 | 248 | 157 | 96 | 35 | 5 | 874 | 813 | 752 | 691 | 630 | 569 | 508 | 447 | 386 | 325 | 264 |
| 232 | 171 | 110 | 49 | 19 | 888 | 827 | 766 | 705 | 644 | 583 | 522 | 461 | 400 | 339 | 278 | 187 | 155 | 64 | 933 | 903 | 842 | 781 | 720 | 659 | 598 | 537 | 476 | 415 | 354 | 293 |
| 261 | 200 | 139 | 78 | 947 | 917 | 856 | 795 | 734 | 673 | 612 | 551 | 490 | 429 | 368 | 307 | 246 | 185 | 94 | 62 | 2 | 871 | 810 | 749 | 688 | 627 | 566 | 505 | 444 | 383 | 322 |
| 290 | 229 | 168 | 107 | 46 | 16 | 885 | 824 | 763 | 702 | 641 | 580 | 519 | 458 | 397 | 336 | 275 | 214 | 153 | 92 | 931 | 930 | 839 | 778 | 717 | 656 | 595 | 534 | 473 | 412 | 351 |
| 319 | 258 | 197 | 136 | 75 | 944 | 914 | 853 | 792 | 731 | 670 | 609 | 548 | 487 | 426 | 365 | 304 | 243 | 182 | 121 | 60 | 30 | 869 | 837 | 746 | 685 | 624 | 563 | 502 | 441 | 380 |
| 348 | 287 | 226 | 165 | 104 | 43 | 13 | 882 | 821 | 760 | 699 | 638 | 577 | 516 | 455 | 394 | 333 | 272 | 211 | 150 | 89 | 958 | 928 | 867 | 776 | 744 | 653 | 592 | 531 | 470 | 409 |
| 377 | 316 | 255 | 194 | 133 | 72 | 941 | 911 | 850 | 789 | 728 | 667 | 606 | 545 | 484 | 423 | 362 | 301 | 240 | 179 | 118 | 57 | 27 | 896 | 835 | 774 | 683 | 651 | 560 | 499 | 438 |
| 406 | 345 | 284 | 223 | 162 | 101 | 40 | 10 | 879 | 818 | 757 | 696 | 635 | 574 | 513 | 452 | 391 | 330 | 269 | 208 | 147 | 86 | 955 | 925 | 864 | 803 | 742 | 681 | 590 | 558 | 467 |
| 465 | 374 | 313 | 252 | 191 | 130 | 69 | 938 | 908 | 847 | 786 | 725 | 664 | 603 | 542 | 481 | 420 | 359 | 298 | 237 | 176 | 115 | 54 | 24 | 893 | 832 | 771 | 710 | 649 | 588 | 497 |
| 495 | 404 | 372 | 281 | 220 | 159 | 98 | 37 | 7 | 876 | 815 | 754 | 693 | 632 | 571 | 510 | 449 | 388 | 327 | 266 | 205 | 144 | 83 | 952 | 922 | 861 | 800 | 739 | 678 | 617 | 556 |
| 524 | 463 | 402 | 311 | 279 | 188 | 127 | 66 | 935 | 905 | 844 | 783 | 722 | 661 | 600 | 539 | 478 | 417 | 356 | 295 | 234 | 173 | 112 | 51 | 21 | 890 | 829 | 768 | 707 | 646 | 585 |
| 553 | 492 | 431 | 370 | 309 | 218 | 186 | 95 | 34 | 4 | 873 | 812 | 751 | 690 | 629 | 568 | 507 | 446 | 385 | 324 | 263 | 202 | 141 | 80 | 949 | 919 | 858 | 797 | 736 | 675 | 614 |
| 582 | 521 | 460 | 399 | 338 | 277 | 216 | 125 | 93 | 932 | 902 | 841 | 780 | 719 | 658 | 597 | 536 | 475 | 414 | 353 | 292 | 231 | 170 | 109 | 48 | 18 | 887 | 826 | 765 | 704 | 643 |
| 611 | 550 | 489 | 428 | 367 | 306 | 245 | 184 | 123 | 32 | 31 | 870 | 809 | 748 | 687 | 626 | 565 | 504 | 443 | 382 | 321 | 260 | 199 | 138 | 77 | 946 | 916 | 855 | 794 | 733 | 672 |
| 640 | 579 | 518 | 457 | 396 | 335 | 274 | 213 | 152 | 91 | 960 | 900 | 868 | 777 | 716 | 655 | 594 | 533 | 472 | 411 | 350 | 289 | 228 | 167 | 106 | 45 | 15 | 884 | 823 | 762 | 701 |
| 669 | 608 | 547 | 486 | 425 | 364 | 303 | 242 | 181 | 120 | 59 | 29 | 898 | 807 | 775 | 684 | 623 | 562 | 501 | 440 | 379 | 318 | 257 | 196 | 135 | 74 | 943 | 913 | 852 | 791 | 730 |
| 698 | 637 | 576 | 515 | 454 | 393 | 332 | 271 | 210 | 149 | 88 | 957 | 927 | 866 | 805 | 714 | 682 | 591 | 530 | 469 | 408 | 347 | 286 | 225 | 164 | 103 | 42 | 12 | 881 | 820 | 759 |
| 727 | 666 | 605 | 544 | 483 | 422 | 361 | 300 | 239 | 178 | 117 | 56 | 26 | 895 | 834 | 773 | 712 | 621 | 589 | 498 | 437 | 376 | 315 | 254 | 193 | 132 | 71 | 940 | 910 | 849 | 788 |
| 756 | 695 | 634 | 573 | 512 | 451 | 390 | 329 | 268 | 207 | 146 | 85 | 954 | 924 | 863 | 802 | 741 | 680 | 619 | 528 | 496 | 405 | 344 | 283 | 222 | 161 | 100 | 39 | 9 | 878 | 817 |
| 785 | 724 | 663 | 602 | 541 | 480 | 419 | 358 | 297 | 236 | 175 | 114 | 53 | 23 | 892 | 831 | 770 | 709 | 648 | 587 | 526 | 435 | 403 | 312 | 251 | 190 | 129 | 68 | 937 | 907 | 846 |
| 814 | 753 | 692 | 631 | 570 | 509 | 448 | 387 | 326 | 265 | 204 | 143 | 82 | 951 | 921 | 860 | 799 | 738 | 677 | 616 | 555 | 494 | 433 | 342 | 310 | 219 | 158 | 97 | 36 | 6 | 875 |
| 843 | 782 | 721 | 660 | 599 | 538 | 477 | 416 | 355 | 294 | 233 | 172 | 111 | 50 | 20 | 889 | 828 | 767 | 706 | 645 | 584 | 523 | 462 | 401 | 340 | 249 | 217 | 126 | 65 | 934 | 904 |
| 872 | 811 | 750 | 689 | 628 | 567 | 506 | 445 | 384 | 323 | 262 | 201 | 140 | 79 | 948 | 918 | 857 | 796 | 735 | 674 | 613 | 552 | 491 | 430 | 369 | 308 | 247 | 156 | 124 | 33 | 3 |
| 901 | 840 | 779 | 718 | 657 | 596 | 535 | 474 | 413 | 352 | 291 | 230 | 169 | 108 | 47 | 17 | 886 | 825 | 764 | 703 | 642 | 581 | 520 | 459 | 398 | 337 | 276 | 215 | 154 | 63 | 961 |
This 31x31 magic square is panmagic and symmetric.
You can use this method to construct ultra magic squares of order is prime number from 5x5 to infinity. See 5x5, 7x7, 11x11, 13x13, 17x17, 19x19, 23x23, 29x29, 31x31
