10x10 and 11x11 magic squares of cubes.
10x10 and 11x11 magic squares of 4th powers.
10x10 and 11x11 magic squares of 5th powers.

In October 2006, some days after the first known 9x9 magic square of cubes, I was pleased to construct the first known 10x10 magic square of cubes:

• it uses the first 100 cubes, from 13 to 1003
• 13, 23, 33, and 43 are placed in the first row

•  143 933 43 13 23 33 943 303 93 963 863 543 53 903 803 263 353 173 653 563 193 133 873 773 233 153 593 913 313 753 213 473 693 273 443 993 463 643 673 723 1003 453 343 283 603 83 533 813 793 203 413 103 893 743 513 613 163 823 683 523 403 783 713 73 383 223 853 183 953 483 333 973 493 923 293 363 573 623 253 583 763 113 63 433 983 883 323 373 633 423 663 123 703 393 833 843 733 503 243 553

And in November 2006, I constructed the first known 11x11 magic square of cubes:

• it uses the first 121 cubes, from 13 to 1213
• all 11 rows remain magic when the entries are not cubed. S1=671.

•  363 843 333 113 633 1003 923 1193 23 653 663 513 343 313 763 243 883 373 143 1153 1053 963 743 573 1063 353 83 1033 323 623 583 203 1163 863 783 223 873 183 103 703 1213 973 393 433 1093 793 643 1133 493 553 253 673 123 33 953 53 213 503 563 693 13 933 723 993 1073 983 593 153 753 1023 713 413 1183 173 193 1013 533 423 1123 853 733 1103 43 943 543 403 133 443 813 283 913 273 1113 1083 603 63 823 613 163 1173 73 303 263 1043 833 463 683 903 773 233 293 1203 1143 893 383 483 523 473 93 803 453

Open problems:

• 10x10 and 11x11 magic squares of 4th powers are unknown.
• 10x10 and 11x11 magic squares of 5th powers are unknown.

March 2018, Nicolas Rouanet, France, constructed this 10x10 (and also a 12x12) nearly-magic square of consecutive 4th powers, from 1^4 to 100^4:

 444 374 184 524 824 874 254 854 734 544 24 774 744 954 74 144 264 344 834 554 354 124 634 174 134 364 924 484 604 994 14 594 904 864 284 314 104 574 884 274 814 1004 534 224 654 464 514 644 474 424 684 624 294 794 454 764 584 394 154 944 894 194 404 164 724 934 334 704 564 414 204 64 914 674 714 304 974 114 324 244 694 804 44 54 964 94 664 234 784 214 844 34 614 384 84 754 494 984 434 504

Nicolas Rouanet remarked, using modulo 5 reasoning, that an 11x11 magic (or semi-magic) square is impossible using consecutive 4th powers, from 1^4 to 121^4.