Problems & Puzzles: Puzzles

 Puzzle 749. Prime magic cubes Natalia Makarova proposes the following puzzleA "magic cube" is the 3-dimensional equivalent of a magic square, that is, a collection of distinct number of integers arranged in an n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic constant of the cube. In a contest recently organized by Natalia and friends, she got the following solution for the case 4x4x4, with a magic constant equal to 780, not being sure if this solution is the minimal magic constant possible for this case (4x4x4): n=4, S=780 (author N. Makarova) 17 7 439 317 139 487 107 47 331 59 167 223 293 227 67 193 -----------------19 61 281 419 191 199 179 211 337 347 43 53 233 173 277 97 ------------------283 443 23 31 421 83 127 149 3 103 433 241 73 151 197 359 ------------------461 269 37 13 29 11 367 373 109 271 137 263 181 229 239 131----------------- Q. Can you produce another 4x4x4 prime magic cube with smaller magic constant than 780?

Natalia Makarova sent the following results on May 1, 2015:

In the contest found the best solutions for puzzle #749

n=5, S=3505 (not minimal, author Michael Hürter)

439 1049 17 1429 571

2467 331 71 277 359

263 1093 607 509 1033

307 419 2137 631 11

29 613 673 659 1531

. . . . . . . . . . . . . . . . .

61 1223 811 587 823

409 751 1123 773 449

569 733 1091 1069 43

1187 757 197 37 1327

1279 41 283 1039 863

. . . . . . . . . . . . . . . . .

1549 347 433 79 1097

239 3 1433 1087 743

599 179 701 809 1217

541 2069 211 547 137

577 907 727 983 311

. . . . . . . . . . . . . . . . .

839 797 761 1061 47

53 619 421 1051 1361

1583 563 97 199 1063

641 229 787 971 877

389 1297 1439 223 157

. . . . . . . . . . . . . . . . .

617 89 1483 349 967

337 1801 457 317 593

491 937 1009 919 149

829 31 173 1319 1153

1231 647 383 601 643

n=6, S=5670 (not minimal, my solution)

971 761 1801 367 157 1613

1447 379 491 1031 569 1753

1033 1667 1709 281 877 103

1117 1777 419 457 1607 293

461 23 787 1723 797 1879

641 1063 463 1811 1663 29

. . . . . . . . . . . . . . . . . . . . . .

607 409 719 1877 1627 431

1543 1039 31 887 1823 347

599 439 2311 383 647 1291

233 353 557 2371 499 1657

1229 1949 881 139 811 661

1459 1481 1171 13 263 1283

. . . . . . . . . . . . . . . . . . . . . .

937 149 1303 677 983 1621

1553 97 307 2069 1307 337

727 2267 523 317 673 1163

653 967 1451 271 1091 1237

1531 449 1499 1123 709 359

269 1741 587 1213 907 953

. . . . . . . . . . . . . . . . . . . . . .

1277 1693 19 1289 829 563

593 53 2333 751 643 1297

1423 991 107 373 2309 467

311 2423 571 659 127 1579

739 313 769 1997 701 1151

1327 197 1871 601 1061 613

. . . . . . . . . . . . . . . . . . . . . .

17 1831 401 1381 1847 193

397 2591 1109 73 7 1493

101 83 839 2707 151 1789

1759 37 1201 479 2063 131

1699 1069 631 521 1559 191

1697 59 1489 509 43 1873

. . . . . . . . . . . . . . . . . . . . . .

1861 827 1427 79 227 1249

137 1511 1399 859 1321 443

1787 223 181 1609 1013 857

1597 113 1471 1433 283 773

11 1867 1103 167 1093 1429

277 1129 89 1523 1733 919

n=7, S=18053 (not minimal, author Michael Hürter)

4889 5659 3373 443 547 263 2879

859 1481 4357 3889 631 3769 3067

941 643 131 2549 5927 3313 4549

997 2663 3733 2633 2153 4967 907

2417 3433 3221 739 4649 457 3137

5867 293 2969 1637 3989 2441 857

2083 3881 269 6163 157 2843 2657

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1931 23 1213 2837 5657 4261 2131

1039 4801 3391 4177 1913 2161 571

5227 3023 4057 1831 823 2239 853

1531 1283 1217 1151 4787 4441 3643

3719 3229 2939 3347 2141 1759 919

2999 1933 2069 4519 1033 1571 3929

1607 3761 3167 191 1699 1621 6007

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4759 4481 487 2311 1061 2711 2243

683 809 2833 109 6917 6473 229

349 4793 3169 4909 673 4099 61

6491 1489 1667 5233 1483 1619 71

4079 1289 2557 1063 5021 2677 1367

401 1861 5563 967 991 277 7993

1291 3331 1777 3461 1907 197 6089

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1009 2423 4861 4423 3499 181 1657

1511 6673 107 2371 19 1523 5849

6529 3911 1709 79 2087 2137 1601

1373 2333 5297 2579 41 3467 2963

2383 1163 271 2143 1873 5653 4567

167 1399 3527 5647 5783 953 577

5081 151 2281 811 4751 4139 839

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 4483 2801 1453 101 3779 5237

5003 1433 2293 1031 2203 937 5153

2351 2713 1847 4789 3251 1783 1319

2213 3109 2671 173 5323 2207 2357

2179 2957 2621 6779 491 2887 139

2749 3329 1693 1109 563 6143 2467

3359 29 4127 2719 6121 317 1381

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

977 227 5107 4349 2797 3299 1297

5839 89 1951 4447 1499 3041 1187

1427 2251 1549 1459 2819 431 8117

251 6389 2447 601 3557 1069 3739

2617 1249 5351 3623 409 2027 2777

2689 6761 239 2273 4111 1237 743

4253 1087 1409 1301 2861 6949 193

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4289 757 211 2237 4391 3559 2609

3119 2767 3121 2029 4871 149 1997

1229 719 5591 2437 2473 4051 1553

5197 787 1021 5683 709 283 4373

659 4733 1093 359 3469 2593 5147

3181 2477 1993 1901 1583 5431 1487

379 5813 5023 3407 557 1987 887

Questions:

1. Find the magic cubes of orders 4 - 7 with smallest magic constant.

2. Find the magic cubes of orders for n>7.

Contest

***

On May 26, 2015, Natalia Makarova wrote:

I got a wonderful solutions from Michael Hürter for the puzzle #749.

This magic cubes of orders 9, 10, 11 and 30.

See attachment.

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