Ultra magic square of 11-th order

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  • Aprile 22, 2015 alle 8:21 am #322
    Natalia Makarova
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    Scheme for ultra magic square of order 11

    The general formula of ultra magic square of order 11

    X(1) = -(-4*X(52)-2*X(55)+3*K-4*X(16)-2*X(8)-4*X(57)-4*X(22)+2*X(23)+4*X(15) +2*X(3)-2*X(26)-4*X(27)-2*X(28)+4*X(29)-2*X(30)+4*X(10)-2*X(58) -4*X(32)-6*X(33)+4*X(34)+2*X(35)-4*X(36)+6*X(4)-2*X(38)+2*X(39) +2*X(40)+6*X(9)+4*X(59)-2*X(43)-4*X(44)+2*X(45)-4*X(46)-4*X(47) +6*X(5)+2*X(7)) /4 
X(11) = -(4*X(53)+4*X(54)+6*X(55)-21*K-4*X(16)-2*X(8)-4*X(57)+4*X(22)+6*X(23) -4*X(15)-2*X(3)-2*X(26)-12*X(27)-2*X(28)+4*X(29)-2*X(30)+12*X(10) -2*X(58)-4*X(32)-2*X(33)+6*X(35)+6*X(4)-6*X(38)-6*X(39)+6*X(40) +2*X(9)+8*X(59)-
4*X(42)+2*X(43)+2*X(45)+8*X(47)+4*X(48)+2*X(5) +4*X(50)+6*X(7)) /4   
X(12) = (-4*X(52)+3*K-4*X(16)-2*X(8)-2*X(57)-2*X(22)+2*X(23)-4*X(27)-2*X(28) +4*X(29)+4*X(10)-4*X(32)-2*X(33)+2*X(34)+2*X(35)-2*X(36)+2*X(4) -2*X(38)+2*X(40)+4*X(9)+4*X(59)-2*X(42)-2*X(43)-2*X(44)+2*X(5) +2*X(50)+2*X(7)) /2 
X(13) = (-2*X(52)-2*X(53)-4*X(54)-2*X(55)+11*K+2*X(16)+2*X(8)-4*X(22)-6*X(23) +4*X(15)+2*X(3)+6*X(27)+2*X(28)-2*X(29)-4*X(10)+2*X(58)-4*X(35) +4*X(38)+4*X(39)-2*X(40)-4*X(59)-2*X(44)-2*X(45)-2*X(46)-6*X(47) +2*X(5)-2*X(50)-2*X(7)+2*X(60)) /2 
X(14) = (-2*X(55)+3*K-4*X(16)+2*X(8)+4*X(57)+2*X(23)-4*X(15)-2*X(3)-2*X(26) -2*X(28)+2*X(30)+2*X(58)+2*X(33)+2*X(35)-6*X(4)-2*X(38)-2*X(39) -2*X(40)+2*X(9)-2*X(43)+2*X(45)+4*X(46)+4*X(47)-2*X(5)+2*X(7)) /4
 X(17) = 2*X(52)+ X(54)+ X(55)-5*K+ X(8)+ X(57)+2*X(22)+ X(23)-2*X(15)- X(3)- X(27)- X(29) +2*X(32)+ X(33)+ X(35)+ X(36)- X(4)- X(39)- X(9)+ X(42)+ X(43)+2*X(44)+ X(45) + X(46)+2*X(47)-2*X(5)- X(60) 
X(18) = - X(52)- X(53)- X(54)- X(55)+6*K+ X(16)-2*X(22)-2*X(23)+2*X(15)+2*X(3)+3*X(27) - X(29)-2*X(10)-2*X(35)+ X(38)+2*X(39)- X(40)-2*X(59)- X(45)- X(46)-3*X(47) - X(48)+ X(5)- X(50)- X(7)+ X(60)
X(19) = (-2*X(52)+4*X(53)+4*X(54)+2*X(55)-7*K-4*X(16)-4*X(8)-4*X(57)+6*X(23)-2*X(15) -2*X(3)-8*X(27)-2*X(28)+4*X(29)+8*X(10)-2*X(58)-4*X(32)-2*X(33) +2*X(34)+4*X(35)-2*X(36)+4*X(4)-4*X(38)-4*X(39)+4*X(40)+2*X(9) +6*X(59)-2*X(42)-2*X(44)+2*X(45)+4*X(47)+2*X(48)+2*X(5)+4*X(50) +2*X(7)) /2  
X(2) = (2*X(54)+ K-2*X(8)-2*X(57)+2*X(23)-2*X(27)+2*X(29)+2*X(10)-2*X(58)-2*X(33) +2*X(4)-2*X(38)+2*X(40)+2*X(59)-2*X(46)) /2 
X(20) = -(-8*X(52)-2*X(55)+ K-4*X(16)-2*X(8)-4*X(57)-8*X(22)+2*X(23)+4*X(15) +2*X(3)-2*X(26)-4*X(27)-2*X(28)+4*X(29)+2*X(30)+8*X(10)-2*X(58) -8*X(32)-6*X(33)+4*X(34)+2*X(35)-4*X(36)+6*X(4)-2*X(38)+2*X(39) +6*X(40)+6*X(9)+4*X(59)-4*X(42)-2*X(43)-4*X(44)+2*X(45)-4*X(46) -4*X(47)+6*X(5)+4*X(50)+2*X(7)+4*X(60)) /4
X(21) = -(-2*X(52)- K-2*X(16)-2*X(28)+2*X(58)+2*X(33)+2*X(34)-2*X(4)-2*X(40) +2*X(9)+2*X(46)) /2 
X(24) = 2*X(52)+ X(53)+2*X(54)+ X(55)-5*K+2*X(22)+2*X(23)-2*X(15)- X(3)-2*X(27)+2*X(10) - X(58)+ X(32)- X(34)+ X(35)- X(38)-2*X(39)+ X(40)- X(9)+ X(59)+ X(43)+ X(44)+ X(45) +3*X(47)- X(5)+ X(50)+ X(7)- X(60)  
X(25) = - X(54)+5*K+ X(16)-2*X(23)+ X(15)+ X(27)- X(29)- X(30)-2*X(10)- X(35)+ X(38)+ X(39) - X(40)- X(59)+ X(42)- X(45)-2*X(47)- X(50)- X(7)
X(31) = (-4*X(52)-2*X(53)-2*X(54)-2*X(55)+11*K-2*X(16)-4*X(22)-2*X(23)+2*X(15) +2*X(3)-2*X(26)-2*X(28)+2*X(58)-4*X(32)-2*X(33)+2*X(34)+2*X(39) +2*X(9)-2*X(42)-2*X(43)-2*X(44)-2*X(47)+2*X(5)+2*X(60)) /2 
X(37) = (4*X(53)+4*X(54)+2*X(55)+5*K-4*X(16)-2*X(8)-4*X(57)+6*X(23)-4*X(15)-2*X(3) -2*X(26)-8*X(27)-2*X(28)+4*X(29)+2*X(30)+8*X(10)-2*X(58)-4*X(32) -2*X(33)+2*X(35)-4*X(36)+2*X(4)-6*X(38)-6*X(39)+2*X(40)+2*X(9) +4*X(59)-4*X(42)-2*X(43)-4*X(44)+2*X(45)+4*X(47)+2*X(5)+4*X(50) +2*X(7)) /4 
X(41) = -(4*X(53)+4*X(54)+2*X(55)-17*K-4*X(16)-2*X(8)-4*X(57)+6*X(23)-4*X(15) -2*X(3)-2*X(26)-8*X(27)-2*X(28)+4*X(29)+2*X(30)+8*X(10)-2*X(58) -4*X(32)-2*X(33)+4*X(34)+6*X(35)+2*X(4)-2*X(38)-2*X(39)+6*X(40) +2*X(9)+4*X(59)+2*X(43)+2*X(45)+4*X(47)+2*X(5)+4*X(50)+2*X(7)) /4
X(49) = - X(52)- X(53)- X(54)- X(55)+6*K- X(22)- X(23)+ X(15)+ X(3)+ X(27)- X(10)- X(35)+ X(39) - X(59)- X(45)- X(46)-2*X(47)- X(48)- X(50)
X(51) = (- K+2*X(22)+2*X(23)-2*X(15)-2*X(3)-2*X(27)+2*X(10)+2*X(35)-2*X(39) +2*X(59)+2*X(47)) /2 
X(56) = X(52)- X(53)- X(54)- X(55)+5*K+2*X(16)+ X(8)+ X(57)-3*X(23)+2*X(15)+ X(3)+4*X(27) + X(28)-2*X(29)-4*X(10)+2*X(32)+ X(33)- X(34)-2*X(35)- X(4)+2*X(38)+2*X(39) -2*X(40)- X(9)-3*X(59)+2*X(42)+ X(44)- X(45)- X(46)-3*X(47)- X(48)-2*X(50) -2*X(7) 
X(6) = (-2*X(52)+2*X(53)+2*X(55)+ K-4*X(16)-2*X(8)-2*X(57)+2*X(23)-2*X(3)-2*X(26) -6*X(27)-2*X(28)+2*X(29)-2*X(30)+4*X(10)-4*X(32)-2*X(33)+2*X(34) +4*X(35)-2*X(36)+2*X(4)-2*X(38)-2*X(39)+2*X(40)+2*X(9)+4*X(59) -2*X(42)-2*X(44)+2*X(45)+2*X(47)+2*X(48)+2*X(5)+2*X(50)+2*X(7)) /2

    We have 40 free variables and 20 dependent variables (if the constant associativity K is set).

    Minimal magic constant for ultra magic square of order 11 is 26521.
    I’m trying to find a minimal solution.
    While I have a solution with 9 errors:

    4709*2441 2423 4229   23 2549  101  431  401 4751 4463
3617 3449 1361  569  809 4787  239 4655*4379*1493 1163
 563 3803 4001 3203 1229  449  149 3761 4061*1913 3389
 179 1979  941 3413*3671 4049  173 4601 2579 3413 1523
2819 3209 1571 3851 1289 3719 4079 4139    5 1811   29
1091  911 3539 4733  131 2411 4691   89 1283 3911 3731*
4793 3011 4817  683  743 1103 3533  971 3251 1613 2003
3299 1409 2243 221  4649  773 1151 1409*3881 2843 4643
1433 2909  761 1061 4673 4373 3593 1619  821 1019 4259
3659 3329  443  167 4583  35* 4013 4253 3461 1373 1205*
 359   71 4421 4391 4721 2273 4799  593 2399  2381 113

    K=4822, S=26521

    Aprile 25, 2015 alle 5:47 am #324
    Natalia Makarova
    Partecipante

    Progress!
    In that solution 7 errors:

    1013 3881 2423 3671 1451 1487 2081  239 1523 4751 4001
4673 2693 1433  569 4013 4577* 101 3449 2441 1409 1163
 563 2711 4463  593 1229   23 2213 3803 3461 3089 4373
1823 1979 1061 1709 4391 4049 1619 1085*4649 1913 2243
2819 3209 1571 3851 1289 3719 4079 4139    5 1811   29
3329  911 3539 4733  131 2411 4691   89 1283 3911 1493
4793 3011 4817  683  743 1103 3533  971 3251 1613 2003
2579 2909  173 3737*3203  773  431 3113*3761 2843 2999
 449 1733 1361 1019 2609 4799 3593 4229  359 2111 4259
3659 3413 2381 1373 4721  245* 809 4253 3389 2129  149
 821   71 3299 4583 2741 3335*3371 1151 2399  941 3809*

    K=4822, S=26521

    Aprile 28, 2015 alle 2:07 pm #326
    Natalia Makarova
    Partecipante

    And more progress!
    In this solution 5 errors

    2711 4463 2423 1409 4673 1913    5  431 1361 4751 2381
2291*2741 3299  569  821 3545*3221 4583 2999 1289 1163
 563 3389 2243 1733 4079  101   23 3323 4493 3881 2693
 593 1979 3761 3203  179 4049 3803 611* 4421 2549 1373
2819 3209 1571 3851  809 3719 1229 4139 1151 1811 2213
3257  911 3539 4733  131 2411 4691   89 1283 3911 1565*
2609 3011 3671  683 3593 1103 4013  971 3251 1613 2003
3449 2273  401 4211 1019  773 4643 1619 1061 2843 4229
2129  941 329* 1499 4799 4721  743 3089 2579 1433 4259
3659 3533 1823  239 1601 1277 4001 4253 1523 2081 2531
2441   71 3461 4391 4817 2909  149 3413 2399  359 2111

    K=4822, S=26521

    Maggio 12, 2015 alle 7:13 pm #342
    Natalia Makarova
    Partecipante

    I tried to find a solution with other magic constant, S = 51073.
    While I have a solution with errors:

    8933 6323 4649  113 3593 7253    5   59 5279 6947 7919 
8969  857 8093 2837 2099 8501  257 8375 7247  449 3389 
4493 6863 3299 5297 2207 3023 1613  419 7043 8273 8543 
3083 4283 3359 5861 8783 6287 7853 7589 2267  659 1049 
3257 2687 3803 6389 1409 5879 9239 6359  773 6869 4409 
1217 6827 4049 6947 5519 4643 3767 2339 5237 2459 8069 
4877 2417 8513 2927   47 3407 7877 2897 5483 6599 6029 
8237 8627 7019 1697 1433 2999  503 3425 5927 5003 6203 
 743 1013 2243 8867 7673 6263 7079 3989 5987 2423 4793 
5897 8837 2039  911 9029  785 7187 6449 1193 8429  317 
1367 2339 4007 9227 9281 2033 5693 9173 4637 2963  353

    K = 9286, S = 51073

    Errors:

    8375 – not prime
    6947 – not unique
    2339 – not unique
    3425 – not prime
    785 – not prime
    2033 – not prime

    Maggio 15, 2015 alle 10:44 am #344
    Natalia Makarova
    Partecipante

    Progress!

    In this solution three errors

    8447 8093 4649  113   47 5357    5  503 8273 6977 8609
8543  857 4013 2837 4133 8699 1709 7877 5669 3347 3389
4493 9257 8969 9059 2963 1637  659  419 4349 5279 3989
1787 4283 2207 2393 8837 6287 7247 7937 3593 4889 1613
3257 2687 3803 6389 1277 5879 8513 6359 1163 6869 4877
 593 6827 4049 6947 5519 4643 3767 2339 5237 2459 8693
4409 2417 8123 2927  773 3407 8009 2897 5483 6599 6029
7673 4397 5693 1349 2039 2999  449 6893 7079 5003 7499
5297 4007 4937 8867 8627 7649 6323  227  317   29 4793
5897 5939 3617 1409 7577  587 5153 6449 5273 8429  743
 677 2309 1013 8783 9281 3929 9239 9173 4637 1193  839

    K = 9286, S = 51073

    Errors:

    5357 – not prime
    1349 – not prime
    6893 – not prime

    Maggio 16, 2015 alle 7:42 am #346
    Natalia Makarova
    Partecipante

    And more progress!
    In this solution there are two errors

    4097 8237 4649 2963 9257 1787    5 7079 2243 6977 3779 
5279 1979 6329 2837   83 7019 9059 8819  293 5987 3389 
4493 7793 1559 8513 6203 3023 1709  149 7919 1013 8699 
2633 4283 4889  617   47 6287 9227 1217 8543 6317 7013 
3257 2687 3803 6389 8807 5879 5273 6359 1637 6869  113 
7877 6827 4049 6947 5519 4643 3767 2339 5237 2459 1409 
9173 2417 7649 2927 4013 3407  479 2897 5483 6599 6029 
2273 2969  743 8069   59 2999 9239 8669 4397 5003 6653 
 587 8273 1367 9137 7577 6263 3083  773 7727 1493 4793 
5897 3299 8993  467  227 2267 9203 6449 2957 7307 4007 
5507 2309 7043 2207 9281 7499   29 6323 4637 1049 5189

    K=9286, S=51073

    4097 is not prime
    8993 is not prime

    Maggio 27, 2015 alle 7:07 am #356
    Natalia Makarova
    Partecipante

    And more progress!

    In the solution there is only one error:

    8477 7187 4649  503 7577 3617    5 6329 4703 6977 1049 
 839  659 5279 2837 9203 7499 8969 3299 8513  587 3389 
4493 6863 7589 1433 6203 2633 9239 2039 2969  593 7019 
5153 4283 3989 6197  227 6287  857 1637 6323 8837 7283 
3257 2687 3803 6389  257 5879 5273 6359 2207 6869 8093 
9257 6827 4049 6947 5519 4643 3767 2339 5237 2459   29 
1193 2417 7079 2927 4013 3407 9029 2897 5483 6599 6029 
2003  449 2963 7649 8429 2999 9059 3089 5297 5003 4133 
2267 8693 6317 7247   47 6653 3083 7853 1697 2423 4793 
5897 8699  773 5987  317 1787   83 6449 4007 8627 8447 
8237 2309 4583 2957 9281 5669 1709 8783 4637 2099  809

    8477 is not prime

    K = 9286, S = 51073

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