Crea sito

Concentric magic squares of primes

Home Forum Primes Magic Games site Concentric magic squares of primes

  • Questo topic ha 15 risposte, 2 partecipanti ed è stato aggiornato l'ultima volta 2 anni, 9 mesi fa da Anna.
Stai visualizzando 15 post - dal 1 a 15 (di 16 totali)
  • Autore
    Post
  • #595
    Natalia Makarova
    Partecipante

    On this web page
    http://www.magic-squares.net/primesqr.htm

    you can see the concentric magic square of order 13 composed of prime numbers

    1153 8923 1093 9127 1327 9277 1063 9133 9661 1693 991 8887 8353
    9967 8161 3253 2857 6823 2143 4447 8821 8713 8317 3001 3271 907
    1831 8167 4093 7561 3631 3457 7573 3907 7411 3967 7333 2707 9043
    9907 7687 7237 6367 4597 4723 6577 4513 4831 6451 3637 3187 967
    1723 7753 2347 4603 5527 4993 5641 6073 4951 6271 8527 3121 9151
    9421 2293 6763 4663 4657 9007 1861 5443 6217 6211 4111 8581 1453
    2011 2683 6871 6547 5227 1873 5437 9001 5647 4327 4003 8191 8863
    9403 8761 3877 4783 5851 5431 9013 1867 5023 6091 6997 2113 1471
    1531 2137 7177 6673 5923 5881 5233 4801 5347 4201 3697 8737 9343
    9643 2251 7027 4423 6277 6151 4297 6361 6043 4507 3847 8623 1231
    1783 2311 3541 3313 7243 7417 3301 6967 3463 6907 6781 8563 9091
    9787 7603 7621 8017 4051 8731 6427 2053 2161 2557 7873 2713 1087
    2521 1951 9781 1747 9547 1597 9811 1741 1213 9181 9883 1987 9721

    S=70681

    I got some interesting results on this issue. I’ll show these results.
    And I talk about the unresolved issues.

    We consider concentric magic squares composed of prime numbers.

    1. I found a minimal concentric magic square of order 5 with magic constant S=1255.

    2. I found a concentric magic square of order 6 with magic constant S=630.

    Maybe it’s not a minimal solution.
    I invite colleagues to investigate this matter.

    To be continued…

    #597
    Natalia Makarova
    Partecipante

    3. I found a minimal concentric magic square of order 7 with magic constant S=4487

    4. I found a concentric magic square of order 8 with magic constant S=2040

    Maybe it’s not a minimal solution.

    To be continued…

    #599
    Natalia Makarova
    Partecipante

    5. I found a minimal concentric magic square of order 9 with magic constant S=12249

    6. I found a minimal concentric magic square of order 10 with magic constant S=4200

    Maybe it’s not a minimal solution.

    Here we have an unsolved problem.
    In sequence OEIS
    https://oeis.org/A188537

    you see: a(10) <= 4950

    I found an associative square of order 10 with magic constant S = 4950

    593 977 757 181 331 307 569 281 193 761
    631 541 953  83 677 131 257 419 641 617
    673 127 383 137 239 727 719 433 911 601
    937 113 103 467 983 827  43 167 499 811
    839 547  23 971 109 347 251 919  61 883
    107 929  71 739 643 881  19 967 443 151
    179 491 823 947 163   7 523 887 877  53
    389  79 557 271 263 751 853 607 863 317
    373 349 571 733 859 313 907  37 449 359
    229 797 709 421 683 659 809 233  13 397

    K=990, S=4950

    But I have not found a minimal associative square of order 10 with magic constant
    S = 4200.
    Can anyone solve this problem?

    To be continued…

    #602
    Natalia Makarova
    Partecipante

    Refinement
    6. I found a concentric magic square of order 10 with magic constant S=4200

    #603
    Natalia Makarova
    Partecipante

    7. I found a minimal concentric magic square of order 11 with magic constant S=26521

    8. I found a concentric magic square of order 12 with magic constant S=8820

    Maybe it’s not a minimal solution.

    There are minimal associative squares of orders 11, 12.
    See https://oeis.org/A188537

    a(11) = 26521, a(12)=8820

    I found these solutions:

    n=11

    4079 4673  401 3911 1151 2111 1619 1229 4583 1601 1163
    2381 1361 4733 4253  941 1523 1823 3803  809 4721  173
     563 1499 1373  179 4001 3371 2579 2213 3539 2741 4463
    3389 1979  431 1493 2069 4049 3413   23 1913 4229 3533
    2819 3209 1571 3851 2399 3719  449 4139    5 1811 2549
    1061 4793 4751  131 1733 2411 3089 4691   71   29 3761
    2273 3011 4817  683 4373 1103 2423  971 3251 1613 2003
    1289  593 2909 4799 1409  773 2753 3329 4391 2843 1433
     359 2081 1283 2609 2243 1451  821 4643 3449 3323 4259
    4649  101 4013 1019 2999 3299 3881  569   89 3461 2441
    3659 3221  239 3593 3203 2711 3671  911 4421  149  743

    K=4822, S=26521

    n=12

     241 1061   11  307  523  317 1429  809  719 1021 1291 1091
    1277 1237  487 1039   37 1427  659  181 1049  503  277  647
     541  587  727  907 1187  163  499 1013 1399  643 1051  103
      31  877 1297 1069  797 1451  761  461  829  211  683  353
     439  191 1231  701  149 1303 1097 1123  857  853  607  269
    1381   17  839 1087 1373  151  367  479  257 1361   61 1447
      23 1409  109 1213  991 1103 1319   97  383  631 1453   89
    1201  863  617  613  347  373  167 1321  769  239 1279 1031
    1117  787 1259  641 1009  709   19  673  401  173  593 1439
    1367  419  827   71  457  971 1307  283  563  743  883  929
     823 1193  967  421 1289  811   43 1433  431  983  233  193
     379  179  449  751  661   41 1153  947 1163 1459  409 1229

    K=1470, S=8820

    To be continued…

    #605
    Natalia Makarova
    Partecipante

    9. I found a minimal concentric magic square of order 13 with magic constant S=49439

    10. I found a concentric magic square of order 14 with magic constant S=16170

    Maybe it’s not a minimal solution.

    Here we have an unsolved problem.
    In sequence OEIS
    https://oeis.org/A188537
    you see: a(13) <= 50453

    I found an associative square of order 13 with magic constant S = 50453

       5 7673 1409 5939  659 7193 5861 2789 7589 1061 5153  179 4943
    5099   59 7283 5021 3533 3083  683 6263 5519 4391 1979  431 7109
    3911   71 6581 5231  113 6899 5009 6599  743 2969 5693 3011 3623
    3089 2411 6269 1559 1913 4919 2381 7499  641 4001 5813 6659 3299
    7001 3389 5651 2459 6389 1193 3833 6311  929 3413 4271 4643  971
    5051  281  719 6053 5171 7253 4091 4721 4799 7529 1433 1733 1619
    1289 6653  311  983 6323 7523 3881  239 1439 6779 7451 1109 6473
    6143 6029 6329  233 2963 3041 3671  509 2591 1709 7043 7481 2711
    6791 3119 3491 4349 6833 1451 3929 6569 1373 5303 2111 4373  761
    4463 1103 1949 3761 7121  263 5381 2843 5849 6203 1493 5351 4673
    4139 4751 2069 4793 7019 1163 2753  863 7649 2531 1181 7691 3851
     653 7331 5783 3371 2243 1499 7079 4679 4229 2741  479 7703 2663
    2819 7583 2609 6701  173 4973 1901  569 7103 1823 6353   89 7757

    K=7762, S=50453

    But I have not found a minimal associative square of order 13 with magic constant
    S = 49439.
    Can anyone solve this problem?

    To be continued…

    #607
    Natalia Makarova
    Partecipante

    11. I found a minimal concentric magic square of order 15 with magic constant S=74595

    12. I found a concentric magic square of order 16 with magic constant S=21840

    Maybe it’s not a minimal solution.

    In sequence OEIS
    https://oeis.org/A188537
    you see: a(15)=74595, a(16)<=25200

    I found a minimal associative square of order 15 with magic constant S=74595

    6983 9137 1277 5807 5273 2357 7109 2153 8573 9437 5309  107 1559 7487 2027
    8963  773  359 6779 4133 9497 3557  947 7523 4289 4049 5939 4253 3593 9941
    3083 5573  653 5399 8753 5717 2039 7529 1997 2069 9923 9629 5867 2447 3917
    9203 8837 2063 2693 1163  977 8423 6857 6977 3833 7193  719 3617 4937 7103
    3623 3119 8849 5297 9059 2243 2399 2339  479 8363 9857 4943 3659 3677 6689
    1709  569 9719 2903 9929 7247 3209 1877 3347 1283 8933 5147 4793 8447 5483
    9677  467 8537 2129 9689  179 8627 2999 2927 3803   59 7673 8039 2663 7127
    4019 9749 5303 8123 2297 9833 1433 4973 8513  113 7649 1823 4643  197 5927
    2819 7283 1907 2273 9887 6143 7019 6947 1319 9767  257 7817 1409 9479  269
    4463 1499 5153 4799 1013 8663 6599 8069 6737 2699   17 7043  227 9377 8237
    3257 6269 6287 5003   89 1583 9467 7607 7547 7703  887 4649 1097 6827 6323
    2843 5009 6329 9227 2753 6113 2969 3089 1523 8969 8783 7253 7883 1109  743
    6029 7499 4079  317   23 7877 7949 2417 7907 4229 1193 4547 9293 4373 6863
       5 6353 5693 4007 5897 5657 2423 8999 6389  449 5813 3167 9587 9173  983
    7919 2459 8387 9839 4637  509 1373 7793 2837 7589 4673 4139 8669  809 2963

    K=9946, S=74595

    For n = 16 we have unsolved problems.
    I found a associative square of order 16 with magic constant S=25200

    1087  619 2089  607 1249  677  397 2659 2857 2213 1543  151 2053 3001 2797 1201
    1481 1063  421 2801  271 2843  773 1549 2027  313 1013 2593 2731 2767 1697  857
     179 1997 3121 2099  683 3019 1483 2221  907  601  223 2741 1571 2749 1163  443
    2953 3089 2551  863 2557 1597  181 1367 1627 1409  109 1823 1723  113  461 2777
    2267 1777 1933 1493 1721  101 2477  479  769 2719 2081  233 1019  839 2609 2683
     193 3119  127 1913 2297 1873  751  139  709  971  881 3083 1831 1871 2503 2939
    2711 1283 1289 1583 2803  947  997 1151 1789  241 3137 2887  811   89 2791  691
     761 2693 2239  463 2663 1009 2579 2423 1303 1171 3079  733   83 2341 1619   41
    3109 1531  809 3067 2417   71 1979 1847  727  571 2141  487 2687  911  457 2389
    2459  359 3061 2339  263   13 2909 1361 1999 2153 2203  347 1567 1861 1867  439
     211  647 1279 1319   67 2269 2179 2441 3011 2399 1277  853 1237 3023   31 2957
     467  541 2311 2131 2917 1069  431 2381 2671  673 3049 1429 1657 1217 1373  883
     373 2689 3037 1427 1327 3041 1741 1523 1783 2969 1553  593 2287  599   61  197
    2707 1987  401 1579  409 2927 2549 2243  929 1667  131 2467 1051   29 1153 2971
    2293 1453  383  419  557 2137 2837 1123 1601 2377  307 2879  349 2729 2087 1669
    1949  353  149 1097 2999 1607  937  293  491 2753 2473 1901 2543 1061 2531 2063

    K=3150, S=25200

    But I have not found a minimal associative square of order 16 with magic constant
    S = 21840.
    Can anyone solve this problem?

    To be continued…

    #610
    Natalia Makarova
    Partecipante

    13. I found a minimal concentric magic square of order 17 with magic constant S=128197

     6569  6581  6653  6659  6719  6911  6959  6971  7001  7013  7229  7253  7433  7499  7559 11069 12119
     2801  5279  5333  5393  5531  5651  5669  5711  5741  5801  5843  5861 12539 12809 12953 13001 12281
     2843  4079  4133  4229  4283  4349  4373  4391  4451  4481  9491 13103 13463 13523 13763 11003 12239
     2879  4871  4493  3449  3461  3533  3593  3659  3671  6113 13709 13781 13931 14051 10589 10211 12203
     3761  4889  4523  2969  2339  2393  2411  2423  2441 12263 14489 14519 14591 12113 10559 10193 11321
     6329  4919  4583  3011  1823     5   191   269  8219 14633 14699 14771 13259 12071 10499 10163  8753
     8609  4931  4691  3041  1931   359   839   929  6053 14831 15053 14723 13151 12041 10391 10151  6473
     8693  4943  4751  3251  4649  8291  7043 13649   113  8861  8039  6791 10433 11831 10331 10139  6389
     8951  5003  4793 11279 13859 14561 14843  2753  7541 12329   239   521  1223  3803 10289 10079  6131
     9203  9929 12143 13499 14033 14621 14951  6221 14969  1433   131   461  1049  1583  2939  5153  5879
     9851 11783 12983 13721 14321 14639    29 14153  9029   251 14243   443   761  1361  2099  3299  5231
    10061 12473 13043 13799 14423   311 14891 14813  6863   449   383 15077   659  1283  2039  2609  5021
    11093 12491 13109 13901   491 12689 12671 12659 12641  2819   593   563 12743  1181  1973  2591  3989
    11159 12503 13469  1031 11621 11549 11489 11423 11411  8969  1373  1301  1151 11633  1613  2579  3923
    11171 12941  1319 10853 10799 10733 10709 10691 10631 10601  5591  1979  1619  1559 10949  2141  3911
    11261  2081  9749  9689  9551  9431  9413  9371  9341  9281  9239  9221  2543  2273  2129  9803  3821
     2963  8501  8429  8423  8363  8171  8123  8111  8081  8069  7853  7829  7649  7583  7523  4013  8513

    http://s29.postimg.org/yz4v4037r/concentr17_Min.png

    14. I found a concentric magic square of order 18 with magic constant S=35910

      991 1019 1021 1399 1433 1607 1609 1613 1619 1657 1693 1877 2269 3023 3167 3181 3359 3373
    1901 1439 1447 1451 1459 1523 1531 1543 1549 1553 1567 1579 1601 3347 3371 3433 3527 2089
    1907 1051 1187 1193 1201 1213 1223 1237 1249 1277 1291 2503 3533 3593 3607 3623 2939 2083
    1951 1171 1063  827  853  881  907  911  929  941 3019 3631 3659 3673 3709 2927 2819 2039
    1973 1297 1087  761  677  683  691  719  733 1327 3739 3761 3767 3853 3229 2903 2693 2017
    1997 1301 1093  769  647  499  521  523  541 2281 3851 3863 3881 3343 3221 2897 2689 1993
    2003 1307 1103  773  659  443   59   61  113 3907 3911 3919 3547 3331 3217 2887 2683 1987
    2111 1319 1129  787  661  449  211   23 3947 3943   67 3779 3541 3329 3203 2861 2671 1879
    2143 1373 1153 1597 2293 3037 3823 1303 2707 2711 1259  167  953 1697 2393 2837 2617 1847
    2179 1381 2011 3571 3697 3793 3889 2731 1279 1283 2687  101  197  293  419 1979 2609 1811
    2203 2957 3511 3617 3719 3797 3917 3923   47   43 3967   73  193  271  373  479 1033 1787
    2207 3169 3529 3637 3727 3833   71 3929 3877   83   79 3931  157  263  353  461  821 1783
    2213 3389 3557 3643 3733  109 3469 3467 3449 1709  139  127 3491  257  347  433  601 1777
    2243 3391 3559 3677  137 3307 3299 3271 3257 2663  251  229  223 3313  313  431  599 1747
    2267 3413 3581  281 3137 3109 3083 3079 3061 3049  971  359  331  317 3163  409  577 1723
    3251 3499  367 2797 2789 2777 2767 2753 2741 2713 2699 1487  457  397  383 2803  491  739
    1753  463 2543 2539 2531 2467 2459 2447 2441 2437 2423 2411 2389  643  619  557 2551 2237
     617 2971 2969 2591 2557 2383 2381 2377 2371 2333 2297 2113 1721  967  823  809  631 2999

    Maybe it’s not a minimal solution.

    I have not found a minimal associative squares of order 17, 18 for sequence OEIS https://oeis.org/A188537
    I ask colleagues to solve this problem.

    To be continued…

    #613
    Natalia Makarova
    Partecipante
    #614
    Natalia Makarova
    Partecipante

    I found a minimal associative square of order 18

     929 1303  419 1901 2143 3343  599 2237 3169 3499 3251 3947 1373 1033   79 3779  193 2713
    3527  643 1567 2797 1301  331 3917 1433  601 3433  139 1619  373 1553 3823 3889 3877 1087
    1283 3919 3137 3257  991  809 1213  263 2281  359 2269 2447 3547 2381 1697  787 2803 2467
     229 3923 2609 2693 3373 3299 1171 3637  971 1237  397 2003  577 2179 3557  223 2393 1439
    3491 1223  313 2699 2731 3167 1777 3037  457 2539 2927 2543 1153  257 2683 3359   23 1531
     461 1129 3217  293 3931  449 1327 2017 3739 3023 1997  317 1487 3623 1879 1021 2441 3559
    1607 1103 3163 3271 3853 1693  659 1201  881  383  911  197 3719 3907 3943 1723 1613 2083
    2129 3229 3313 3371 1601 2039 1019  661  683 1249 3469  541 2207 3929 3863 1319  941  347
    3581  523 2243 1399 1051 1093 3511 2333 3709  769 1459 1579 1787 2113  157 3881 2711 2011
    1979 1279  109 3833 1877 2203 2411 2531 3221  281 1657  479 2897 2939 2591 1747 3467  409
    3643 3049 2671  127   61 1783 3449  521 2741 3307 3329 2971 1951 2389  619  677  761 1861
    1907 2377 2267   47   83  271 3793 3079 3607 3109 2789 3331 2297  137  719  827 2887 2383
     431 1549 2969 2111  367 2503 3673 1993  967  251 1973 2663 3541   59 3697  773 2861 3529
    2459 3967  631 1307 3733 2837 1447 1063 1451 3533  953 2213  823 1259 1291 3677 2767  499
    2551 1597 3767  433 1811 3413 1987 3593 2753 3019  353 2819  691  617 1297 1381   67 3761
    1523 1187 3203 2293 1609  443 1543 1721 3631 1709 3727 2777 3181 2999  733  853   71 2707
    2903  113  101  167 2437 3617 2371 3851  557 3389 2557   73 3659 2689 1193 2423 3347  463
    1277 3797  211 3911 2957 2617   43  739  491  821 1753 3391  647 1847 2089 3571 2687 3061

    K=3990, S=35910

    #616
    Natalia Makarova
    Partecipante

    I found a minimal associative square of order 17

     3761 13499  8501  2843  2879  1433  4451  4373 12143  5021 12659  9551  8819  6791  9281  9239 12953
     8171    29  7583  6569  6971 14969  9689  9029 10691 12941  5669  8363 13799  2591  4871  4649  1613
    10151 11261  3911   491 10139  6659 14723   461  8081  8783  5879 10601  1049  8069  7829 13151  6959
    13901  3251  1301  3011 11633  7649 13931 13463  2969  2753 12671   251 11423  3989 14321  9341  2339
     3671 14639  9851   383 10289  5591 12503  7523  1823 12281  8609  3299  8969  1973  8039  8951  9803
     4751 14891 13721   521  4691  2963  1031 13001  2039 14423 14153 13859  3803  4079  3461  5861 10949
    10079  1319 12983 12041   563 12641  6221 12809  2393 13709  4493  4919  1979  4229 10193  5153 12473
    14489  4013  4283 14243  5711   269 14771  6863   449  3593 15077  2543  3923  7229 11549  4349 14843
    10499  5333 14951  6389  8429  8753  9431 13523  7541  1559  5651  6329  6653  8693   131  9749  4583
      239 10733  3533  7853 11159 12539     5 11489 14633  8219   311 14813  9371   839 10799 11069   593
     2609  9929  4889 10853 13103 10163 10589  1373 12689  2273  8861  2441 14519  3041  2099 13763  5003
     4133  9221 11621 11003 11279  1223   929   659 13043  2081 14051 12119 10391 14561  1361   191 10331
     5279  6131  7043 13109  6113 11783  6473  2801 13259  7559  2579  9491  4793 14699  5231   443 11411
    12743  5741   761 11093  3659 14831  2411 12329 12113  1619  1151  7433  3449 12071 13781 11831  1181
     8123  1931  7253  7013 14033  4481  9203  6299  7001 14621   359  8423  4943 14591 11171  3821  4931
    13469 10433 10211 12491  1283  6719  9413  2141  4391  6053  5393   113  8111  8513  7499 15053  6911
     2129  5843  5801  8291  6263  5531  2423 10061  2939 10709 10631 13649 12203 12239  6581  1583 11321

    K=15082, S=128197

    To be continued…

    #617
    Natalia Makarova
    Partecipante

    15. I found a minimal concentric magic square of order 19 with magic constant S=191159

     8699  8849  8951  8963  9029  9173  9239  9323  9413  9431  9521  9533  9689  9833  9851  9929 12203 13001 16529
     3191  7211  7331  7433  7451  7481  7583  7643  7649  7793  7841  7853  7883  9791 17159 17573 17579 17783 16931
     3491  4493  5231  6323  6359  6473  6569  6599  6653  6659  6701  6791 13163 18119 18251 18503 18521 15629 16631
     4349  4649  4973  5309  5351  5381  5399  5483  5501  5531  5573 12893 18413 18539 18671 18749 15149 15473 15773
     7079  4679  5021  5711  3803  3821  3929  4019  4049  4241 10499 18959 19031 19139 19181 14411 15101 15443 13043
     8111  4721  5039  5801  3221  2633  2729  2789  2801  2939 18143 19403 19553 19559 16901 14321 15083 15401 12011
     8219  4793  5153  5879  3251  2063     5    59   149 10631 19841 19853 19889 18059 16871 14243 14969 15329 11903
    10181  7001  5171  6089  3299  2081   359   821   863  8861 19739 20021 19763 18041 16823 14033 14951 13121  9941
    11549  7013  6863  6113  3359  6263 11369  9341 18311   173 11699 10781  8753 13859 16763 14009 13259 13109  8573
    11579  7019  6971  6221 17099 18803 19379 19991  3449 10061 16673   131   743  1319  3023 13901 13151 13103  8543
    11621 15641 15233 14843 18713 19211 19421 20051  8423 19949  1811    71   701   911  1409  5279  4889  4481  8501
    11831 16421 17681 17981 18899 19463 19661   101 19259 11261   383 19301   461   659  1223  2141  2441  3701  8291
    12041 16451 17729 18233 19013 19469   233 20063 19973  9491   281   269 20117   653  1109  1889  2393  3671  8081
    12113 16661 17909 18401 19073   563 17393 17333 17321 17183  1979   719   569 17489  1049  1721  2213  3461  8009
    15671 17033 18149 18839   941 16301 16193 16103 16073 15881  9623  1163  1091   983 16319  1283  1973  3089  4451
    15731 17123 18191  1373 14771 14741 14723 14639 14621 14591 14549  7229  1709  1583  1451 14813  1931  2999  4391
    15749 17789  1601 13799 13763 13649 13553 13523 13469 13463 13421 13331  6959  2003  1871  1619 14891  2333  4373
    16361  2339 12791 12689 12671 12641 12539 12479 12473 12329 12281 12269 12239 10331  2963  2549  2543 12911  3761
     3593 11273 11171 11159 11093 10949 10883 10799 10709 10691 10601 10589 10433 10289 10271 10193  7919  7121 11423

    http://s9.postimg.org/5fcsqcv9r/Concentr19_Nin.png

    To be continued…

    #618
    Natalia Makarova
    Partecipante

    16. I found a concentric magic square of order 20 with magic constant S=54600

    1361 1367 1381 1409 1433 1439 1447 2543 2551 2557 2609 2617 2657 2659 4133 4139 4229 4597 4673 4799
    1103 2273 2293 2297 2341 2351 2377 2381 2393 2399 2411 2423 2437 2441 2459 2503 3989 4651 4721 4357
    1123 1733 1759 1901 1931 1933 1949 1993 1997 1999 2003 2011 2027 2339 4919 4951 4969 4999 3727 4337
    1163 1741 2053 1783 1787 1789 1801 1823 1847 1867 1877 1879 1889 4691 5051 5059 5077 3407 3719 4297
    1171 2141 2069 1493 1543 1549 1553 1571 1579 1583 1597 1607 4733 5113 5153 5179 3967 3391 3319 4289
    1187 2153 2087 1531 1627 1039 1051 1063 1069 1087 1097 5107 5227 5237 5323 3833 3929 3373 3307 4273
    1201 2161 2089 1609 1637 1283  557  571  599  643 3547 5279 5297 5347 4177 3823 3851 3371 3299 4259
    1217 2203 2099 1613 1657 1301  467   19   41  157 5333 5413 5417 4993 4159 3803 3847 3361 3257 4243
    1229 2207 2113 1693 1663 1303  487  251   11 5443 5437   29 5209 4973 4157 3797 3767 3347 3253 4231
    2663 2239 2129 1699 1667 1453 4441 5309   79 5387 5393   61  151 1019 4007 3793 3761 3331 3221 2797
    4153 2243 2131 1721 2269 5167 5233 5351 5399   67   73 5381  109  227  293 3191 3739 3329 3217 1307
    4201 2251 2137 4621 5081 5189 5261 5407 5431   23   17 5449   53  199  271  379  839 3323 3209 1259
    4447 2521 3089 5003 5087 5197 5281   43 5419 5303  127   47 5441  179  263  373  457 2371 2939 1013
    4451 4001 4783 5011 5101 5231  113 4889 4861 4817 1913  181  163 4903  229  359  449  677 1459 1009
    4463 4211 4787 5021 5147  137 4409 4397 4391 4373 4363  353  233  223 4421  313  439  673 1249  997
    4639 4751 4937 5039  281 3911 3907 3889 3881 3877 3863 3853  727  347  307 3917  421  523  709  821
    4663 4759 4957  383 3673 3671 3659 3637 3613 3593 3583 3581 3571  769  409  401 3677  503  701  797
    4703 4813  461 3559 3529 3527 3511 3467 3463 3461 3457 3449 3433 3121  541  509  491 3701  647  757
    4801  739 3167 3163 3119 3109 3083 3079 3067 3061 3049 3037 3023 3019 3001 2957 1471  809 3187  659
     661 4093 4079 4051 4027 4021 4013 2917 2909 2903 2851 2843 2803 2801 1327 1321 1231  863  787 4099

    http://s23.postimg.org/j21uvtw4r/concentr20.png

    Maybe it’s not a minimal solution.

    I found also a minimal associative square of order 20 with magic constant S=54600

    4523 1327 2609 3581   67  199 3727  787 4649 5107  127 4937 5381 3361   11 1283 4969 4651  113 3191
    2503  977 4813 1597 1069 5197 1231 3547 1993 5237  401 1741 5417 5281 4363  677 2767  701 1901 3187
    1087 3037 2069 1123 4441  157 5231 5437 4957 2789 1847 2011  359 4357 1949 1999 1933 5297   73 4447
     383  347 4451 2341 2203 1627 3929 2137 1801 2711 1259 3847 1307 2843 5051 4597 5419 2683 2207 3457
    4507 1187 1171 4297 4639 1609 4861 1867 4259 3167 4463 4013 3739 1877 2161 1543 3001  487 1303  449
    5441 5309 1439 1931 2239 2243 1229  509 2719 2339 4799 1553 1699 3089 2551 4583  947  181 4999 4801
     757 4493 1759 2393 2437 4007 1783 4621   61 1249 5443 1571 1471 4051 2053 3853 1063 4001 2801 4733
     769 1493 1657 4663 2129 3877 4243 2377 2113 1667 3797 5167 5003 3371 5039  739 3671 2153  439  233
    5081  911 4889 4751 2543 3209 4139 1039  227  673 5413  271  557 2707  307 3673 2131 2297 5179 4603
    5431 4093 1693 2657 1823 4409 1997 3079 2441 5407 2797  109  467 1433 3049  643 3433 4099 4159 1381
    4079 1301 1361 2027 4817 2411 4027 4993 5351 2663   53 3019 2381 3463 1051 3637 2803 3767 1367   29
     857  281 3163 3329 1787 5153 2753 4903 5189   47 4787 5233 4421 1321 2251 2917  709  571 4549  379
    5227 5021 3307 1789 4721  421 2089  457  293 1663 3793 3347 3083 1217 1583 3331  797 3803 3967 4691
     727 2659 1459 4397 1607 3407 1409 3989 3889   17 4211 5399  839 3677 1453 3023 3067 3701  967 4703
     659  461 5279 4513  877 2909 2371 3761 3907  661 3121 2741 4951 4231 3217 3221 3529 4021  151   19
    5011 4157 4973 2459 3917 3299 3583 1721 1447  997 2293 1201 3593  599 3851  821 1163 4289 4273  953
    2003 3253 2777   41  863  409 2617 4153 1613 4201 2749 3659 3323 1531 3833 3257 3119 1009 5113 5077
    1013 5387  163 3527 3461 3511 1103 5101 3449 3613 2671  503   23  229 5303 1019 4337 3391 2423 4373
    2273 3559 4759 2693 4783 1097  179   43 3719 5059  223 3467 1913 4229  263 4391 3863  647 4483 2957
    2269 5347  809  491 4177 5449 2099   79  523 5333  353  811 4673 1733 5261 5393 1879 2851 4133  937

    K=5460, S=54600

    #619
    Natalia Makarova
    Partecipante

    I have not found minimal associative square of order 19 with magic constant S = 191159.
    I have only associative square of order 19 with magic constant
    S = 193363.

     6133  6547 14401  2503 11071  2647 16111  1213 20143  8971 18691   883 15361  1117 19501 14713   577 13513 19267
     7591  8053  3391  3907  2137 19597  5167 14947 16993 14923    13 14431 20047  9181  3607 12043 11197  6271 19867
    17293  5113  3163  8887 20023  4363 19183 19681  6673 17041  5527  3613 13873  4021  1831  8803  5503 10111 18661
    14731 16411   241 20347 11113  5077 12163 17467  7867  8101  7177 17047 18553  1381  3511  9601  8467 13297   811
     5791 17383 13003 18133  1597   463 18061 11677  2131  5557 19447 17317  2803  3433  2617 18493 13033 12637  9787
    15331 14011  2593 12613    67  2053 19753 15073  8317  7573  4441  9697 19813 11593  7933 19531  2857  8623 11491
     2311  9397 16657 20173  6763 14281 17053 16477  4447  8521 11617 14107 15451  5323  2341 13693  1987 11827   937
    15493 10723  2143  8011 14143  7753 11287  3217 17623  5821   877 17683 20161 15787 15733  5851   283 11311  9463
     7207   751  1567  9001  7963 19387 17977  2383  5233  9643 13933 14347 16057 17401 13327  9901 16981  1873  8431
    17011 11251 13903 19231  5701  3793 15541 17167  2473 10177 17881  3187  4813 16561 14653  1123  6451  9103  3343
    11923 18481  3373 10453  7027  2953  4297  6007  6421 10711 15121 17971  2377   967 12391 11353 18787 19603 13147
    10891  9043 20071 14503  4621  4567   193  2671 19477 14533  2731 17137  9067 12601  6211 12343 18211  9631  4861
    19417  8527 18367  6661 18013 15031  4903  6247  8737 11833 15907  3877  3301  6073 13591   181  3697 10957 18043
     8863 11731 17497   823 12421  8761   541 10657 15913 12781 12037  5281   601 18301 20287  7741 17761  6343  5023
    10567  7717  7321  1861 17737 16921 17551  3037   907 14797 18223  8677  2293 19891 18757  2221  7351  2971 14563
    19543  7057 11887 10753 16843 18973  1801  3307 13177 12253 12487  2887  8191 15277  9241     7 20113  3943  5623
     1693 10243 14851 11551 18523 16333  6481 16741 14827  3313 13681   673  1171 15991   331 11467 17191 15241  3061
      487 14083  9157  8311 16747 11173   307  5923 20341  5431  3361  5407 15187   757 18217 16447 16963 12301 12763
     1087  6841 19777  5641   853 19237  4993 19471  1663 11383   211 19141  4243 17707  9283 17851  5953 13807 14221

    K=20354, S=193363

    #620
    Natalia Makarova
    Partecipante

    This concentric magic square of order 6 composed of distinct prime numbers is minimal.

    S = 504

    Inside this square we have a magic square of order 4 – not associative!

Stai visualizzando 15 post - dal 1 a 15 (di 16 totali)
  • Devi essere connesso per rispondere a questo topic.

Comments are closed.