Problems & Puzzles: Puzzles

Puzzle 777. Ultra magic squares Natalia Makarova sent the following nice puzzle:   Definition Magic square is called ultra magic square, if it is both associative (center symmetric) and pandiagonal.   Ultra magic squares exist for orders n > 4.   n=5, S=3505 (minimal, my solution)   113 1151 1229 911 101 839 521 41 1013 1091 941 953 701 449 461 311 389 1361 881 563 1301 491 173 251 1289   n=6, S=990 (minimal, author M. Alekseyev)   103 59 163 233 139 293 229 257 307 131 13 53 283 17 67 173 181 269 61 149 157 263 313 47 277 317 199 23 73 101 37 191 97 167 271 227   n=7,  S=4613 (not minimal ? author N. Makarova)   227 617 677 431 1217 1307 137 1259 827 1061 509 521 167 269 347 929 1187 17 557 719 857 89 479 29 659 1289 839 1229 461 599 761 1301 131 389 971 1049 1151 797 809 257 491 59 1181 11 101 887 641 701 1091   n=8, S=2640 (not minimal ? my solution)   137 97 647 229 607 503 281 139 419 311 199 601 83 7 463 557 433 367 347 383 389 631 43 47 397 479 61 173 307 113 467 643 17 193 547 353 487 599 181 263 613 617 29 271 277 313 293 227 103 197 653 577 59 461 349 241 521 379 157 53 431 13 563 523   n=9, S=24237 (not minimal ? author A. Chernov)   5381 5189 5273 149 107 89 83 2633 5333 977 449 443 419 5003 5039 5147 5153 1607 1583 4787 3413 4877 653 1373 3089 2909 1553 2699 3863 743 4127 2027 3767 1979 2609 2423 2969 1709 3119 3389 2693 1997 2267 3677 2417 2963 2777 3407 1619 3359 1259 4643 1523 2687 3833 2477 2297 4013 4733 509 1973 599 3803 3779 233 239 347 383 4967 4943 4937 4409 53 2753 5303 5297 5279 5237 113 197 5   Q1. Is it possible to find ultra magic squares of order 7 - 9 at less magic constant? Q2. Required to find the ultra magic squares of order n > 9.

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Natalia Makarova wrote on April 22

I found a solution to the puzzle # 777.

n=8, S=2040 (minimal)

 

241 199 409 467 47 79 359 239

421 137 7 53 487 179 317 439

31 281 347 353 227 277 127 397

449 197 109 379 491 337 11 67

443 499 173 19 131 401 313 61

113 383 233 283 157 163 229 479

71 193 331 23 457 503 373 89

271 151 431 463 43 101 311 269

See https://oeis.org/A257316

I invite colleagues for the contest on this problem

http://primesmagicgames.altervista.org/wp/competitions/

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On July 13, 2015 Natalia wrote:

 

I found a new solutions to the puzzle # 777.

n=9, S=13059 (not minimal ?)

2843 149 1973 2039 971 1031 2141 293 1619
2063 563 1811 113 2549 1601 2633 1721 5
2393 503 1613 2381 1193 41 2411 101 2423
173 2711 2879 773 1583 1493 461 443 2543
569 83 821 311 1451 2591 2081 2819 2333
359 2459 2441 1409 1319 2129 23 191 2729
479 2801 491 2861 1709 521 1289 2399 509
2897 1181 269 1301 353 2789 1091 2339 839
1283 2609 761 1871 1931 863 929 2753 59

n=10, S=46150 (not minimal)

9133 2017 1069 1669 3583 4999 8629 1489 6343 7219
5209 4219 5101 6793 43 6841 7951 2683 5557 1753
7603 7369 6883 8059 8863 919 1471 769 4111 103
163 8179 4723 4243 4663 5869 1741 6553 1723 8293
2833 4051 709 1021 7177 5701 1993 6991 8101 7573
1657 1129 2239 7237 3529 2053 8209 8521 5179 6397
937 7507 2677 7489 3361 4567 4987 4507 1051 9067
9127 5119 8461 7759 8311 367 1171 2347 1861 1627
7477 3673 6547 1279 2389 9187 2437 4129 5011 4021
2011 2887 7741 601 4231 5647 7561 8161 7213 97


See
http://primesmagicgames.altervista.org/wp/result-for-ultra-magic-squares-of-prime-numbers/

 

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