3x3 Magic Squares with Six Squared Integers Frequency of magic numbers within a given magnitude +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | | X | | | X | X | | X | | X | | | X | | | | | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | X | X | | X | | X | | X | | X | | X | X | X | | X | X | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | X | X | X | | X | X | | | X | | X | | X | | X | | X | X | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ 10^1 0 10^1 0 10^1 0 10^1 0 10^1 0 10^2 0 10^2 0 10^2 0 10^2 0 10^2 0 10^3 0 10^3 0 10^3 0 10^3 0 10^3 0 10^4 0 10^4 1 10^4 0 10^4 0 10^4 2 10^5 2 10^5 0 10^5 0 10^5 0 10^5 4 10^6 1 10^6 7 10^6 3 10^6 3 10^6 9 10^7 3 10^7 16 10^7 12 10^7 7 10^7 31 10^8 9 10^8 22 10^8 6 10^8 4 10^8 47 10^9 9 10^9 58 10^9 9 10^9 32 10^9 88 10^10 21 10^10 158 10^10 45 10^10 39 10^10 239 10^11 29 10^11 220 10^11 99 10^11 67 10^11 286 10^12 61 10^12 364 10^12 129 10^12 125 10^12 530 10^13 94 10^13 939 10^13 315 10^13 285 10^13 1300 10^14 116 10^14 1293 10^14 234 10^14 325 10^14 1748 10^15 154 10^15 2026 852 10^15 481 10^15 2632 10^16 184 10^16 4160 10^16 1027 10^16 3256 10^17 28 10^17 803 10^17 1181 10^17 27 10^18 36 10^18 50 10^18 1336 10^18 38 10^19 70 10^19 61 10^19 2440 10^19 65 10^20 70 10^20 46 10^20 2599 10^20 65 10^21 94 10^21 38 10^21 3107 10^21 84 10^22 176 10^22 60 10^22 4592 10^22 153 10^23 180 10^23 59 10^23 3945 10^23 154 10^24 258 10^24 76 10^24 4013 10^24 198 10^25 52 10^25 139 10^25 4896 10^25 361 1647 10^26 140 10^26 2874 10^26 366 10^27 182 10^27 2411 10^27 411 10^28 329 10^28 2647 10^28 457 10^29 334 10^29 1605 10^29 434 10^30 432 10^30 1317 10^30 564 10^31 561 10^31 855 10^31 730 10^32 1016 10^32 722 10^32 1322 10^33 1029 10^33 131 10^33 1339 10^34 1334 10^34 1 10^34 1738 10^35 171 57857 10^35 3144 16124 10^36 3185 10^37 4132 10^38 6975 10^39 1703 37817 +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | X | | | | | X | | | | X | | | | X | | | | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | X | X | X | | X | X | X | | X | X | | | X | | X | | X | X | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | X | X | | X | | X | | X | X | X | | X | X | X | | | X | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ 10^1 0 10^1 0 10^1 0 10^1 0 10^1 0 10^2 0 10^2 0 10^2 0 10^2 0 10^2 0 10^3 0 10^3 0 10^3 0 10^3 0 10^3 0 10^4 0 10^4 2 10^4 1 10^4 1 10^4 1 10^5 2 10^5 4 10^5 2 10^5 3 10^5 4 10^6 2 10^6 6 10^6 2 10^6 1 10^6 7 10^7 10 10^7 35 10^7 12 10^7 26 10^7 12 10^8 11 10^8 52 10^8 20 10^8 28 10^8 26 10^9 21 10^9 82 10^9 31 10^9 47 10^9 39 10^10 62 10^10 216 10^10 66 10^10 117 10^10 114 10^11 80 10^11 250 10^11 73 10^11 162 10^11 119 10^12 108 10^12 468 10^12 122 10^12 209 10^12 177 10^13 220 10^13 972 10^13 229 10^13 493 10^13 335 10^14 271 10^14 1178 10^14 267 10^14 518 10^14 392 10^15 377 10^15 1832 10^15 367 10^15 705 10^15 544 10^16 800 10^16 3874 10^16 459 10^16 818 10^16 756 10^17 585 10^17 3622 10^17 29 10^17 22 10^17 67 10^18 375 10^18 3584 1680 10^18 1 10^18 1 10^19 713 10^19 5862 3151 2594 10^20 587 10^20 4964 10^21 686 10^21 5158 10^22 943 10^22 6806 10^23 696 10^23 4932 10^24 705 10^24 4416 10^25 828 10^25 5354 10^26 572 10^26 3276 10^27 448 10^27 2812 10^28 538 10^28 3142 10^29 264 10^29 1726 10^30 204 10^30 1224 10^31 119 10^31 866 10^32 92 10^32 578 10^33 13 10^33 124 10332 67417 +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | | X | | | X | | | | X | X | | X | | X | | | X | X | | X | | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | X | X | X | | X | | X | | X | X | | | | X | | | X | | | | | | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | X | X | | | X | X | X | | X | | X | | X | X | X | | X | X | X | | X | X | X | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ 10^1 0 10^1 0 10^1 0 10^1 0 10^1 0 10^1 0 10^2 0 10^2 0 10^2 0 10^2 0 10^2 0 10^2 0 10^3 0 10^3 0 10^3 0 10^3 0 10^3 1 10^3 0 10^4 0 10^4 0 10^4 2 10^4 1 10^4 2 10^4 0 10^5 2 10^5 2 10^5 5 10^5 2 10^5 4 10^5 5 10^6 5 10^6 2 10^6 12 10^6 3 10^6 4 10^6 2 10^7 12 10^7 10 10^7 30 10^7 18 10^7 28 10^7 10 10^8 27 10^8 13 10^8 52 10^8 26 10^8 42 10^8 21 10^9 31 10^9 20 10^9 84 10^9 41 10^9 74 10^9 22 10^10 82 10^10 61 10^10 296 10^10 108 10^10 136 10^10 52 10^11 92 10^11 69 10^11 415 10^11 125 10^11 166 10^11 89 10^12 142 10^12 107 10^12 701 10^12 234 10^12 291 10^12 105 10^13 318 10^13 230 10^13 1676 10^13 486 10^13 628 10^13 242 10^14 373 10^14 258 10^14 2222 10^14 589 10^14 777 10^14 297 10^15 562 10^15 391 10^15 3563 10^15 916 10^15 1086 10^15 379 10^16 1090 10^16 818 10^16 6192 10^16 1937 10^16 2172 10^16 467 10^17 904 10^17 666 10^17 248 10^17 1811 10^17 1878 10^17 56 10^18 726 10^18 623 10^18 66 10^18 1792 10^18 1601 1747 10^19 1147 10^19 907 10^19 103 10^19 2931 10^19 2309 10^20 983 10^20 682 10^20 77 10^20 2482 10^20 1810 10^21 962 10^21 661 10^21 80 10^21 2579 10^21 1788 10^22 1220 10^22 812 10^22 140 10^22 3403 10^22 2214 10^23 912 10^23 558 10^23 141 10^23 2466 10^23 1567 10^24 795 10^24 499 10^24 181 10^24 2208 10^24 1298 10^25 1084 10^25 554 10^25 330 10^25 2677 10^25 1498 10^26 649 10^26 329 10^26 334 10^26 1638 10^26 938 10^27 583 10^27 281 10^27 432 10^27 1406 10^27 757 10^28 647 10^28 272 10^28 784 10^28 1571 10^28 802 10^29 343 10^29 152 10^29 792 10^29 863 10^29 436 10^30 286 10^30 106 10^30 1030 10^30 612 10^30 345 10^31 171 10^31 69 10^31 1334 10^31 433 10^31 221 10^32 128 10^32 65 10^32 2416 10^32 289 10^32 159 10^33 22 10^33 11 10^33 2447 10^33 62 10^33 38 14298 9228 10^34 1309 33709 10^34 2 32247 25072 report generated on: Sat, 03 Apr 2021 06:58:19 -0400 total number of magic squares: 296229 number of magic squares containing three negative numbers: 0