Multimagic square
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In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth power for 1 ≤ k ≤ P. Thus, a magic square is bimagic if it is 2-multimagic, and trimagic if it is 3-multimagic.
The first 4-magic square, of order 512, was constructed in May 2001 by André Viricel and Christian Boyer; about one month later, in June 2001, Viricel and Boyer presented the first 5-magic square, of order 1024. They also presented a 4-magic square of order 256 in January 2003, and another 5-magic square, of order 729, was constructed in June 2003 by Chinese mathematician Li Wen.
The smallest known normal satanic square, shown below, has order 8.
| 5 | 31 | 35 | 60 | 57 | 34 | 8 | 30 |
| 19 | 9 | 53 | 46 | 47 | 56 | 18 | 12 |
| 16 | 22 | 42 | 39 | 52 | 61 | 27 | 1 |
| 63 | 37 | 25 | 24 | 3 | 14 | 44 | 50 |
| 26 | 4 | 64 | 49 | 38 | 43 | 13 | 23 |
| 41 | 51 | 15 | 2 | 21 | 28 | 62 | 40 |
| 54 | 48 | 20 | 11 | 10 | 17 | 55 | 45 |
| 36 | 58 | 6 | 29 | 32 | 7 | 33 | 59 |
This magic square has a magic constant of 260. Raising every number to the second power yields the following magic square with a sum of 11180.
| 25 | 961 | 1225 | 3600 | 3249 | 1156 | 64 | 900 |
| 361 | 81 | 2809 | 2116 | 2209 | 3136 | 324 | 144 |
| 256 | 484 | 1764 | 1521 | 2704 | 3721 | 729 | 1 |
| 3969 | 1369 | 625 | 576 | 9 | 196 | 1936 | 2500 |
| 676 | 16 | 4096 | 2401 | 1444 | 1849 | 169 | 529 |
| 1681 | 2601 | 225 | 4 | 441 | 784 | 3844 | 1600 |
| 2916 | 2304 | 400 | 121 | 100 | 289 | 3025 | 2025 |
| 1296 | 3364 | 36 | 841 | 1024 | 49 | 1089 | 3481 |
