Ultra magic square 9th order

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  • #306
    Natalia Makarova
    Partecipante

    The scheme ultra magic square 9th order:

    x1 x2 x3 x4 x5 x6 x7 x8 x9
    x10 x11 x12 x13 x14 x15 x16 x17 x18
    x19 x20 x21 x22 x23 x24 x25 x26 x27
    x28 x29 x30 x31 x32 x33 x34 x35 x36
    x37 x38 x39 x40 k/2 k-x40 k-x39 k-x38 k-x37
    k-x36 k-x35 k-x34 k-x33 k-x32 k-x31 k-x30 k-x29 k-x28
    k-x27 k-x26 k-x25 k-x24 k-x23 k-x22 k-x21 k-x20 k-x19
    k-x18 k-x17 k-x16 k-x15 k-x14 k-x13 k-x12 k-x11 k-x10
    k-x9 k-x8 k-x7 k-x6 k-x5 k-x4 k-x3 k-x2 k-x1

    Here k is constant associativity of square, S = 9k/2, if S – magic constant square.

    The general formula of ultra magic square 9th order

    X(1) = (2*X(32)-2*X(34)+2*X(16)-2*X(18)- K+2*X(24)+2*X(8)-2*X(26)+2*X(40))/2 
    X(11) = - X(32)- X(6)- X(34)+ X(35)-2*X(36)+2*X(10)- X(16)- X(12)-2*X(18)- X(7)+ X(20)-2*X(2) - X(22)+4*K- X(26)- X(27)+ X(28)- X(29)+2*X(30)+2*X(40) 
    X(14) = - X(32)- X(6)- X(16)- X(12)- X(2)- X(22)+5*K- X(24)- X(8)- X(4) 
    X(15) = 3*X(32)+3*X(6)+ X(34)- X(35)+3*X(36)-4*X(10)- X(13)+2*X(16)+2*X(12)+2*X(18)+2*X(7) -2*X(20)+4*X(2)+3*X(22)-9*K+ X(24)+2*X(8)+2*X(26)+2*X(27)- X(28) + X(29)-3*X(30)+ X(4)-3*X(40) 
    X(17) = (-2*X(32)-2*X(6)-2*X(36)+2*X(10)-2*X(16)-2*X(12)-2*X(18)-2*X(7)+2*X(20) -2*X(2)-2*X(22)+9*K-2*X(8)-2*X(26)-2*X(27)+2*X(30)+2*X(40)) /2 
    X(19) = -(-2*X(6)-4*X(34)+2*X(35)-4*X(36)+6*X(10)-6*X(18)-2*X(7)+4*X(20)+2*X(3) -2*X(2)-2*X(22)-3*K+2*X(24)+2*X(8)-4*X(26)-2*X(27)+4*X(28) -2*X(29)+4*X(30)+2*X(4)+6*X(40)) /2 
    X(21) = 2*X(32)+2*X(6)- X(35)+ X(36)-2*X(10)- X(13)+2*X(16)+ X(12)+2*X(7)- X(20)+2*X(2)+ X(22) -4*K+ X(24)+2*X(8)-2*X(30)+ X(4)- X(40) 
    X(23) = - X(32)- X(6)+ X(34)+ X(35)+ X(10)- X(16)- X(12)+ X(18)- X(7)- X(3)- X(2)- X(22)+3*K - X(24)- X(8)+ X(29)+ X(30)- X(4) 
    X(25) = - X(32)-2*X(6)-3*X(34)+ X(35)-3*X(36)+4*X(10)+ X(13)- X(16)-4*X(18)-2*X(7)+2*X(20) +2*X(3)-2*X(2)-2*X(22)+4*K-3*X(26)-2*X(27)+2*X(28)-2*X(29) +3*X(30)+ X(4)+4*X(40) 
    X(31) = X(32)+2*X(6)- X(35)+ X(36)-2*X(10)- X(13)+ X(16)+ X(12)+ X(18)+ X(7)- X(20)+2*X(2)+ X(22) -2*K+ X(24)+ X(8)+ X(26)+ X(27)- X(28)-2*X(30)-2*X(40) 
    X(33) = (-4*X(32)-4*X(6)-2*X(34)-4*X(36)+4*X(10)+2*X(13)-2*X(16)-2*X(12)-2*X(18)-2*X(7) +2*X(20)-4*X(2)-2*X(22)+13*K-2*X(24)-2*X(8)-2*X(26)-2*X(27) -2*X(29)+2*X(30)+4*X(40)) /2 
    X(37) = - X(6)- X(34)+ X(35)- X(36)+ X(10)- X(16)+ X(12)- X(18)- X(7)+ X(20)+ X(3)- X(26)+ X(28)- X(29) + X(30)+ X(4)+ X(40) 
    X(38) = X(34)+ X(36)- X(10)+ X(18)- X(20)+ K+ X(26)- X(28)- X(30)- X(40) 
    X(39) = (-6*X(32)-8*X(6)-4*X(34)+4*X(35)-8*X(36)+12*X(10)+4*X(13)-4*X(16)-4*X(12)-8*X(18) -6*X(7)+6*X(20)+2*X(3)-8*X(2)-6*X(22)+17*K-2*X(24)-4*X(8) -6*X(26)-4*X(27)+4*X(28)-4*X(29)+8*X(30)+10*X(40)) /2 
    X(5) = -(4*X(32)+2*X(6)-2*X(34)-2*X(10)+2*X(16)+2*X(12)-2*X(18)+2*X(7)-2*X(20) +2*X(3)+4*X(2)+2*X(22)-9*K+2*X(24)+4*X(8)-2*X(26)-2*X(30) +2*X(4)) /2 
    X(9) = -(-2*X(32)+2*X(10)-2*X(12)+2*X(20)-2*X(2)-2*X(22)- K+2*X(30)+2*X(40))/2

    We have 24 free variables (if the constant associative is set) and 16 dependent variables.

    According to this formula I found the following solution with 4 errors:

    2767 1429 919  3727 883  757  3607 1039 3889
    223  3499 1609 1567 3877 2647 853  2239 2503
    3457 283  3769 2467 307  3583*451* 1471 3229
    3217 1117 1753 1933 1789 3853 1669 2143 1543
    3613 2677 643  3259 2113 967  3583 1549 613
    2683 2083 2557 373  2437 2293 2473 3109 1009
    997  2755 3775*643* 3919 1759 457  3943 769
    1723 1987 3373 1579 349  2659 2617 727  4003
    337  3187 619  3469 3343 499  3307 2797 1459

    K=4226, S=19017

    #307
    Natalia Makarova
    Partecipante

    Ah! There are still a couple of wrong: (1471, 2755).
    It is also a mistake.

    #308
    Natalia Makarova
    Partecipante

    Progress!

    4463 1259 1181 4079 4703 2099 1511 2213 1811
    3461 1163 4583 461  2939 4091 3701 1979 941
    1559 3989 4799 449  971  929  3041 3209 4373
    173  1319 29   5099 3659 3629*2549 2789 4073
    4133 5051 2801 3251 2591 1931 2381 131  1049
    1109 2393 2633 1553 1523 83   5153 3863 5009
    809  1973 2141 4253 4211 4733 383  1193 3623
    4241 3203 1481 1091 2243 4721 599  4019 1721
    3371 2969 3671 3083 479  1103 4001 3923 719

    K=5182, S=23319

    In this solution, there is only one element of the wrong – 3629 (not prime number).
    I’m not mistaken?

    #309
    Natalia Makarova
    Partecipante

    There is a solution with magic constant 23319!

    #327
    Natalia Makarova
    Partecipante

    There is a solution with magic constant 21969!

    Minimal magic constant for ultra magic square of order 9 is 12249.
    Now I’m trying to find a solution to this magic constant.
    We have only one array of primes to produce ultra magic square of order 9 with magic constant 12249

    11 23 29 59 89 101 113 131 173 179 191 263 281 311 383 389 449 479 509 569 593 641 653 659 683 719 743 773 809 821 
    911 1013 1103 1109 1151 1163 1223 1229 1283 1289 1361 1433 1439 1493 1499 1559 1571 1613 1619 1709 1811 1901 1913 1949 1979 
    2003 2039 2063 2069 2081 2129 2153 2213 2243 2273 2333 2339 2411 2441 2459 2531 2543 2549 2591 2609 2621 2633 2663 2693 2699 2711

    Perhaps there is a solution.

    #329
    Natalia Makarova
    Partecipante

    I found a minimal solution for ultra magic square of order 9 with 3 errors:

      59  101 2699  719 1949 2543 2039  911 1229
    1619 1061 2339 1613 1499  659  653  113 2693
    2531  569  311 2081 1163 2591 1013   11 1979
    1913  449  509 1901  173 1289 2333 2243 1439
    2579 2459 1541*2129 1361  593 1181  263  143*
    1283  479  389 1433 2549  821 2213 2273  809
     743 2711 1709  131 1559  641 2411 2153  191
      29 2609 2069 2063 1223 1109  383 1661*1103
    1493 1811  683  179  773 2003   23 2621 2663

    K=2722, S=12249

    I found a minimal associative square of order 9 of distinct primes

    1283  311 1811 2213 1571  569 2039 1163 1289
     773  653 2243 1619 2063  593 2693  383 1229
    1979 1499 2699  641  821   89  809 2003 1709
    1613 2531  101  131 2333 2441 2663  263  173
     113  179 2711  449 1361 2273   11 2543 2609
    2549 2459   59  281  389 2591 2621  191 1109
    1013  719 1913 2633 1901 2081   23 1223  743
    1493 2339   29 2129  659 1103  479 2069 1949
    1433 1559  683 2153 1151  509  911 2411 1439

    K=2722, S=12249

    See
    http://oeis.org/A188537

    #331
    Natalia Makarova
    Partecipante

    Progress!

    In this solution two errors

     191   101  1223  2711   773  2411  2243   263  2333 
    2663  2273   743   593  2543  1163    89  2153    29 
    2039   653  1151  2549  2339   911   521    23  2063 
    1013  2081   113   821  1289  1109  1229  2591  2003 
    1883* 1283  2213   281  1361  2441   509  1439   839 
     719   131  1493  1613  1433  1901  2609   641  1709 
     659  2699  2201* 1811   383   173  1571  2069   683 
    2693   569  2633  1559   179  2129  1979   449    59 
     389  2459   479   311  1949    11  1499  2621  2531

    K=2722, S=12249

    #335
    Natalia Makarova
    Partecipante

    I found a solution with magic constant S = 19269.
    Now I’m trying to find a solution with magic constant S = 18729.
    I have a few solutions with one error.

    For example

    2549   89 3989  383 1193 3323 2543 1409 3251 
    1499  701 4079 2609 2903 1433 3851  503 1151 
    4049 2099  569 3803 1949  941 3929   71 1319 
    1373 3923 3209 1823   29  863  401 3719 3389 
    1721  971  959*  23 2081 4139 3203 3191 2441 
     773  443 3761 3299 4133 2339  953  239 2789 
    2843 4091  233 3221 2213  359 3593 2063  113 
    3011 3659  311 2729 1259 1553   83 3461 2663 
     911 2753 1619  839 2969 3779  173 4073 1613

    K = 4162, S = 18729

    #336
    Natalia Makarova
    Partecipante

    There is a solution with magic constant S = 18729 !

    #337
    Natalia Makarova
    Partecipante

    Now I’m trying to find a solution with magic constant S = 16407.
    We have the following array of 101 primes to produce this ultra magic square:

    23 29 53 89 107 113 179 197 233 239 257 317 347 389 443 479 509 557 563 
    647 677 683 719 743 809 827 857 947 953 983 1013 1097 1103 1187 1223 1229 
    1289 1307 1373 1409 1433 1439 1493 1559 1583 1607 1619 1667 1697 1733 1823 
    1913 1949 1979 2027 2039 2063 2087 2153 2207 2213 2237 2273 2339 2357 2417 
    2423 2459 2543 2549 2633 2663 2693 2699 2789 2819 2837 2903 2927 2963 2969 
    2999 3083 3089 3137 3167 3203 3257 3299 3329 3389 3407 3413 3449 3467 3533 3539 3557 3593 3617 3623

    I suggest using this pattern of residues modulo 5

    3 3 3 3 2 4 4 2 3 
    2 2 3 3 4 2 4 4 3 
    2 2 2 4 2 4 4 3 4 
    2 3 3 3 4 2 4 2 4 
    3 4 3 2 3 4 3 2 3 
    2 4 2 4 2 3 3 3 4 
    2 3 2 2 4 2 4 4 4 
    3 2 2 4 2 3 3 4 4 
    3 4 2 2 4 3 3 3 3

    I will write a program, using the general formula and the pattern.

    #339
    Natalia Makarova
    Partecipante

    I found several solutions with one error.
    For example:

    1013  113  743 2543 1697 3449 3299  857 2693 
    2837 1187 3203 2423 3329  557  509 2339   23 
    3557 2207 2087 2549  647 1289 1979   53 2039 
     827 3083  233 2963  389 3467  239 2237 2969 
    1313* 719 1583  107 1823 3539 2063 2927 2333 
     677 1409 3407  179 3257  683 3413  563 2819 
    1607 3593 1667 2357 2999 1097 1559 1439   89 
    3623 1307 3137 3089  317 1223  443 2459  809 
     953 2789  347  197 1949 1103 2903 3533 2633

    K=3646, S=16407

    #340
    Natalia Makarova
    Partecipante

    I tried to find a solution with magic constant S = 13059.
    In this solution is the one error

     839 1931   59 1319 1709 2633 2459 1871  239 
     659 2621 2309 1973 1949 1889  503  863  293 
    2381 1619  761 1601 2333   41 1493  101 2729 
     773  149 2789   71 353 1289 2423 2393 2819 
    2879  359 2411 2339 1451  563  491 2543   23 
      83  509  479 1613 2549 2831* 113 2753 2129 
     173 2801 1409 2861  569 1301 2141 1283  521 
    2609 2039 2399 1013  953  929  593  281 2243 
    2663 1031  443  269 1193 1583 2843  971 2063

    K=2902, S=13059

    #341
    Natalia Makarova
    Partecipante

    There is a solution with magic constant S = 14967 !
    Now I’m trying to find a solution with magic constants 13997, 13059 and 12249.

    #343
    Natalia Makarova
    Partecipante

    I found a solution S = 13977 with one error:

     263  353  443 2003 2957 2909 2969  797 1283 
    2837 1667 2543 1583 1889  167  809 2459   23 
    1787 2657 2267 2789  557 1019 1709   83 1109 
     977 1733  773 1613  419 2447 1229 1697 3089 
    1193  179 2243  107 1553 2999  863 2927 1913 
      17 1409 1877  659 2687 1493 2333 1373 2129 
    1997 3023 1397*2087 2549  317  839  449 1319 
    3083  647 2297 2939 1217 1523  563 1439  269 
    1823 2309  137  197  149 1103 2663 2753 2843

    K = 3106, S = 13977

    #355
    Natalia Makarova
    Partecipante

    There is a solution with magic constant S = 13977!

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