Ultra magic square of 12-th order

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  • #347
    Natalia Makarova
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    The scheme for ultra magic square of order 12

    The general formula of ultra magic square of order 12

    X(1) = (2*X(64)+2*X(65)+2*X(66)+ X(67)- X(23)-2*X(25)- X(34)-2*X(35)-2*X(37)- X(45)-2*X(46) -2*X(47)- X(49)+ X(50)+ X(51)+ X(52)+ X(53)+ X(54)+ X(55)-2*X(13)- X(57)- X(58) - X(59)+ X(60)+ X(12)+2*X(62)+2*X(63)+2*X(72)) /2 
    X(10) = -(2*X(65)+ X(67)+2*X(5)+ X(23)+2*X(68)+2*X(7)-2*X(25)+2*X(26)+2*X(3)+2*X(4) +2*X(29)-2*X(30)+2*X(31)+ X(34)+2*X(36)-2*X(38)-18*K-2*X(41) +2*X(70)- X(45)+2*X(16)- X(49)+ X(50)- X(51)+ X(52)+ X(53)+2*X(71)- X(54) +3*X(55)+ X(57)+ X(58)+ X(59)+3*X(60)+3*X(12)+2*X(72)) /2 
    X(11) = -( X(67)+ X(23)+2*X(7)+2*X(3)+2*X(4)-2*X(30)- X(34)-2*X(38)-2*X(41)-2*X(42) +2*X(15)- X(45)-2*X(46)- X(49)- X(50)+ X(51)- X(52)- X(53)+2*X(71)- X(54)+ X(55) - X(57)- X(58)+ X(59)+ X(60)+ X(12)) /2 
    X(14) = -(2*X(65)+ X(67)-2*X(5)- X(23)+2 r5+4*X(7)+4*X(26)+4*X(3)+4*X(4)+2*X(29) -2*X(30)+2*X(32)- X(34)-2*X(37)-4*X(38)+6*K-6*X(41)-4*X(42) +2*X(15)-2*X(44)-3*X(45)-4*X(46)-4*X(47)-3*X(49)- X(50)+ X(51)- X(52) -3*X(53)+2*X(71)-3*X(54)+ X(55)-3*X(57)-3*X(58)- X(59)+ X(60)+3*X(12) +4*X(62)) /2 
    X(17) = -(2*X(65)+ X(67)+2*X(5)+ X(23)+2*X(68)+2*X(7)-4*X(25)+4*X(3)+4*X(4)+2*X(29) -2*X(30)-2*X(32)-2*X(33)- X(34)-2*X(37)-4*X(38)-6*K-2*X(41) +2*X(15)-2*X(44)-3*X(45)-2*X(46)+4*X(16)- X(49)- X(50)- X(51)+ X(52) + X(53)+2*X(71)- X(54)+3*X(55)- X(57)- X(58)+ X(59)+3*X(60)+3*X(12) +2*X(72)) /2 
    X(18) = X(64)- X(68)- X(7)- X(25)-2*X(26)-2*X(29)- X(30)-2*X(31)- X(32)-2*X(33)-2*X(69) -2*X(34)- X(35)-2*X(36)+ X(38)+9*K+ X(41)+ X(15)-2*X(70)- X(44)- X(45)- X(46) - X(47)+ X(16)+2*X(49)+ X(50)+ X(51)+ X(52)+ X(53)- X(71)+2*X(54)- X(13)- X(57) - X(58)- X(12)+ X(63) 
    X(19) = -( X(67)+ X(23)-2*X(26)-2*X(4)- X(34)-6*K+2*X(41)+ X(45)+ X(49)+ X(50)+ X(51) + X(52)+ X(53)+ X(54)+ X(55)+ X(57)+ X(58)+ X(59)- X(60)- X(12)+2*X(63)) /2 
    X(2) = X(65)- X(5)+ X(68)+ X(7)+ X(26)+ X(3)+ X(4)+ X(29)+ X(32)+ X(35)- X(37)-2*X(38)+3*K -2*X(41)- X(42)- X(44)- X(45)- X(46)- X(47)- X(49)- X(50)- X(53)+ X(71)- X(54)- X(57) - X(58)+ X(12)+ X(62) 
    X(20) = - X(67)- X(23)-2*X(68)- X(7)+2*X(25)-2*X(3)-2*X(4)+2*X(30)+ X(33)+ X(34)+2*X(38) +2*X(41)-2*X(15)+ X(45)+ X(46)-2*X(16)+ X(49)+ X(50)+ X(51)+ X(52)+ X(53) -2*X(71)+ X(54)- X(55)+ X(57)+ X(58)- X(60)- X(12)- X(72) 
    X(21) = (2*X(65)+ X(67)- X(23)+4*X(68)+4*X(7)+6*X(26)+2*X(3)+4*X(4)+4*X(29)+4*X(31) +2*X(32)+2*X(33)+4*X(69)+3*X(34)+2*X(35)+4*X(36)-2*X(37)-4*X(38) -12*K-6*X(41)-2*X(42)-2*X(15)+4*X(70)- X(45)-5*X(49)-3*X(50) -3*X(51)- X(52)-3*X(53)+4*X(71)-7*X(54)+ X(55)+2*X(13)- X(57)+ X(58)- X(59) + X(60)+5*X(12)+2*X(62)-2*X(63)+2*X(72)) /2 
    X(22) = (2*X(65)+3*X(67)+ X(23)+4*X(68)+4*X(7)-4*X(25)+6*X(3)+4*X(4)+2*X(29)-4*X(30) -2*X(33)-3*X(34)-2*X(37)-6*X(38)-4*X(41)-2*X(42)+4*X(15)-2*X(44) -3*X(45)-4*X(46)-2*X(47)+2*X(16)-3*X(49)- X(50)+ X(51)- X(52)- X(53) +4*X(71)- X(54)+3*X(55)-2*X(13)- X(57)-3*X(58)+ X(59)+3*X(60)+3*X(12) +2*X(62)+2*X(63)+2*X(72)) /2 
    X(24) = (-2*X(64)+ X(67)+ X(23)+2*X(68)+2*X(7)-2*X(25)+4*X(3)+2*X(4)+2*X(29)-2*X(30) - X(34)-4*X(38)-2*X(41)+2*X(15)- X(45)-2*X(46)+2*X(16)- X(49)- X(50) - X(51)- X(52)- X(53)+2*X(71)- X(54)+3*X(55)- X(57)- X(58)+ X(59)+ X(60) + X(12)) /2 
    X(27) = X(64)- X(5)- X(68)- X(25)-2*X(26)- X(29)- X(30)-2*X(31)- X(32)-2*X(33)-2*X(69) -2*X(34)- X(35)-2*X(36)+9*K+ X(40)+ X(41)+ X(15)-2 r3- X(44)- X(45)- X(46) - X(47)+ X(49)+ X(50)+2*X(51)+ X(52)+ X(53)- X(71)+2*X(54)- X(13)- X(57)- X(58) - X(12)+ X(62)+ X(63)- X(72) 
    X(28) = - X(64)+ X(5)+ X(68)+ X(26)+ X(31)+ X(33)+2*X(69)+ X(34)+ X(36)-3*K- X(40)- X(41) - X(15)+2*X(70)+ X(44)+ X(45)+ X(46)+ X(47)- X(49)- X(50)-2*X(51)- X(52)- X(53) + X(71)-2*X(54)+ X(13)+ X(57)+ X(58)+ X(12)- X(62)- X(63)+ X(72) 
    X(39) = - X(64)+ X(67)+2*X(68)+ X(7)+ X(26)+ X(3)+ X(4)+ X(29)+ X(31)+ X(32)+ X(33)+2*X(69)+ X(34) + X(35)+ X(36)- X(37)-2*X(38)- X(40)-2*X(41)- X(42)+2*X(70)-2*X(49)-2*X(50) -2*X(51)-2*X(52)-2*X(53)+2*X(71)-2*X(54)+ X(12)- X(63)+ X(72) 
    X(43) = X(23)- X(25)- X(26)+ X(3)- X(30)- X(31)- X(32)- X(33)- X(69)- X(34)- X(38)+ X(15)- X(70) - X(44)- X(45)- X(46)+ X(16)+ X(49)+ X(50)+ X(51)+ X(52)+ X(53)+ X(54)+ X(55)+ X(59) + X(60)+ X(63) 
    X(48) = X(64)- X(67)- X(23)-2*X(68)- X(7)+ X(25)-2*X(3)- X(4)- X(29)+ X(30)- X(69)- X(35)- X(36) +2*X(38)+6*K+ X(41)- X(15)- r3- X(47)- X(16)+ X(49)+ X(50)+ X(51)+ X(52) + X(53)-2*X(71)+ X(54)- X(55)- X(59)- X(60)- X(12)- X(72) 
    X(36) = 6*K- X(49)- X(50)- X(51)- X(52)- X(53)- X(54)- X(55)- X(57)- X(58)- X(59)- X(60) 
    X(6) = (-2*X(64)-2*X(66)+ X(67)+ X(23)+2*X(68)+4*X(7)+4*X(26)+2*X(3)+2*X(4)+4*X(29) -2*X(30)+4*X(31)+2*X(33)+2*X(69)+3*X(34)+2*X(35)+4*X(36)-4*X(38) -12*K-4*X(41)-2*X(42)+2*X(70)- X(45)+2*X(47)-3*X(49)- X(50)- X(51) - X(52)- X(53)+2*X(71)-5*X(54)+3*X(55)+2*X(13)+ X(57)+ X(58)+ X(59)+3*X(60) +3*X(12)-2*X(63)) /2 
    X(61) = - X(64)- X(65)- X(66)- X(67)- X(68)- X(69)+6*K- X(70)- X(71)- X(62)- X(63)- X(72) 
    X(8) = ( X(67)+ X(23)-2*X(30)- X(34)+2*X(37)-6*K+2*X(15)+ X(45)+ X(49)+ X(50)+ X(51) - X(52)+ X(53)+2*X(71)+ X(54)+ X(55)+ X(57)+ X(58)+ X(59)+ X(60)- X(12)) /2 
    X(9) = -(2*X(65)+ X(67)-2*X(5)- X(23)+2*X(68)+4*X(7)+4*X(26)+2*X(3)+2*X(4)+4*X(29) +2*X(31)+2*X(32)+2*X(33)+2*X(69)+ X(34)+2*X(35)+2*X(36)-2*X(37) -4*X(38)-6*K-4*X(41)-2*X(42)-2*X(44)- X(45)-2*X(46)-2*X(47) -2*X(16)-3*X(49)- X(50)+ X(51)- X(52)- X(53)+2*X(71)-3*X(54)+ X(55)- X(57) - X(58)- X(59)+ X(60)+3*X(12)+4*X(62)) /2

    Where K – a associative constant of square, S = 6K.
    We have 50 free variables of 72 available variables, if given the associative constant K.

    The minimum potential magic constant of ultra magic square of order 12 composed a different prime numbers is 8820.
    Theoretically, the minimal ultra magic square can be made of the following primes

    11 17 19 23 31 37 41 43 47 61 71 89 97 103 109 149 151 163 167 173 179 181 191 193 211 233 239 241 257 269 277 283 307 317 
    347 353 367 373 379 383 401 409 419 421 431 439 449 457 461 479 487 499 503 523 541 563 587 593 607 613 617 631 641 643 647 
    659 661 673 683 701 709 719 727 743 751 761 769 787 797 809 811 823 827 829 839 853 857 863 877 883 907 929 947 967 971 983 
    991 1009 1013 1021 1031 1039 1049 1051 1061 1069 1087 1091 1097 1103 1117 1123 1153 1163 1187 1193 1201 1213 1229 1231 1237 
    1259 1277 1279 1289 1291 1297 1303 1307 1319 1321 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459

    I found a minimal associative square of order 12 composed of the numbers of the array:

     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

    See
    http://oeis.org/A188537

    Perhaps make a ultra magic square of order 12 composed of the numbers of the array?

    #350
    Natalia Makarova
    Partecipante

    I made some errors in the general formula of the ultra magic square of order 12.
    I have now corrected the error and repeat the formula again:

    X(1) = (2*X(64)+2*X(65)+2*X(66)+ X(67)- X(23)-2*X(25)- X(34)-2*X(35)-2*X(37)- X(45)-2*X(46) -2*X(47)- X(49)+ X(50)+ X(51)+ X(52)+ X(53)+ X(54)+ X(55)-2*X(13)- X(57)- X(58) - X(59)+ X(60)+ X(12)+2*X(62)+2*X(63)+2*X(72)) /2 
    X(10) = -(2*X(65)+ X(67)+2*X(5)+ X(23)+2*X(68)+2*X(7)-2*X(25)+2*X(26)+2*X(3)+2*X(4) +2*X(29)-2*X(30)+2*X(31)+ X(34)+2*X(36)-2*X(38)-18*K-2*X(41) +2*X(70)- X(45)+2*X(16)- X(49)+ X(50)- X(51)+ X(52)+ X(53)+2*X(71)- X(54) +3*X(55)+ X(57)+ X(58)+ X(59)+3*X(60)+3*X(12)+2*X(72)) /2 
    X(11) = -( X(67)+ X(23)+2*X(7)+2*X(3)+2*X(4)-2*X(30)- X(34)-2*X(38)-2*X(41)-2*X(42) +2*X(15)- X(45)-2*X(46)- X(49)- X(50)+ X(51)- X(52)- X(53)+2*X(71)- X(54)+ X(55) - X(57)- X(58)+ X(59)+ X(60)+ X(12)) /2 
    X(14) = -(2*X(65)+ X(67)-2*X(5)- X(23)+2*X(68)+4*X(7)+4*X(26)+4*X(3)+4*X(4)+2*X(29) -2*X(30)+2*X(32)- X(34)-2*X(37)-4*X(38)+6*K-6*X(41)-4*X(42) +2*X(15)-2*X(44)-3*X(45)-4*X(46)-4*X(47)-3*X(49)- X(50)+ X(51)- X(52) -3*X(53)+2*X(71)-3*X(54)+ X(55)-3*X(57)-3*X(58)- X(59)+ X(60)+3*X(12) +4*X(62)) /2 
    X(17) = -(2*X(65)+ X(67)+2*X(5)+ X(23)+2*X(68)+2*X(7)-4*X(25)+4*X(3)+4*X(4)+2*X(29) -2*X(30)-2*X(32)-2*X(33)- X(34)-2*X(37)-4*X(38)-6*K-2*X(41) +2*X(15)-2*X(44)-3*X(45)-2*X(46)+4*X(16)- X(49)- X(50)- X(51)+ X(52) + X(53)+2*X(71)- X(54)+3*X(55)- X(57)- X(58)+ X(59)+3*X(60)+3*X(12) +2*X(72)) /2 
    X(18) = X(64)- X(68)- X(7)- X(25)-2*X(26)-2*X(29)- X(30)-2*X(31)- X(32)-2*X(33)-2*X(69) -2*X(34)- X(35)-2*X(36)+ X(38)+9*K+ X(41)+ X(15)-2*X(70)- X(44)- X(45)- X(46) - X(47)+ X(16)+2*X(49)+ X(50)+ X(51)+ X(52)+ X(53)- X(71)+2*X(54)- X(13)- X(57) - X(58)- X(12)+ X(63) 
    X(19) = -( X(67)+ X(23)-2*X(26)-2*X(4)- X(34)-6*K+2*X(41)+ X(45)+ X(49)+ X(50)+ X(51) + X(52)+ X(53)+ X(54)+ X(55)+ X(57)+ X(58)+ X(59)- X(60)- X(12)+2*X(63)) /2 
    X(2) = X(65)- X(5)+ X(68)+ X(7)+ X(26)+ X(3)+ X(4)+ X(29)+ X(32)+ X(35)- X(37)-2*X(38)+3*K -2*X(41)- X(42)- X(44)- X(45)- X(46)- X(47)- X(49)- X(50)- X(53)+ X(71)- X(54)- X(57) - X(58)+ X(12)+ X(62) 
    X(20) = - X(67)- X(23)-2*X(68)- X(7)+2*X(25)-2*X(3)-2*X(4)+2*X(30)+ X(33)+ X(34)+2*X(38) +2*X(41)-2*X(15)+ X(45)+ X(46)-2*X(16)+ X(49)+ X(50)+ X(51)+ X(52)+ X(53) -2*X(71)+ X(54)- X(55)+ X(57)+ X(58)- X(60)- X(12)- X(72) 
    X(21) = (2*X(65)+ X(67)- X(23)+4*X(68)+4*X(7)+6*X(26)+2*X(3)+4*X(4)+4*X(29)+4*X(31) +2*X(32)+2*X(33)+4*X(69)+3*X(34)+2*X(35)+4*X(36)-2*X(37)-4*X(38) -12*K-6*X(41)-2*X(42)-2*X(15)+4*X(70)- X(45)-5*X(49)-3*X(50) -3*X(51)- X(52)-3*X(53)+4*X(71)-7*X(54)+ X(55)+2*X(13)- X(57)+ X(58)- X(59) + X(60)+5*X(12)+2*X(62)-2*X(63)+2*X(72)) /2 
    X(22) = (2*X(65)+3*X(67)+ X(23)+4*X(68)+4*X(7)-4*X(25)+6*X(3)+4*X(4)+2*X(29)-4*X(30) -2*X(33)-3*X(34)-2*X(37)-6*X(38)-4*X(41)-2*X(42)+4*X(15)-2*X(44) -3*X(45)-4*X(46)-2*X(47)+2*X(16)-3*X(49)- X(50)+ X(51)- X(52)- X(53) +4*X(71)- X(54)+3*X(55)-2*X(13)- X(57)-3*X(58)+ X(59)+3*X(60)+3*X(12) +2*X(62)+2*X(63)+2*X(72)) /2 
    X(24) = (-2*X(64)+ X(67)+ X(23)+2*X(68)+2*X(7)-2*X(25)+4*X(3)+2*X(4)+2*X(29)-2*X(30) - X(34)-4*X(38)-2*X(41)+2*X(15)- X(45)-2*X(46)+2*X(16)- X(49)- X(50) - X(51)- X(52)- X(53)+2*X(71)- X(54)+3*X(55)- X(57)- X(58)+ X(59)+ X(60) + X(12)) /2 
    X(27) = X(64)- X(5)- X(68)- X(25)-2*X(26)- X(29)- X(30)-2*X(31)- X(32)-2*X(33)-2*X(69) -2*X(34)- X(35)-2*X(36)+9*K+ X(40)+ X(41)+ X(15)-2*X(70)- X(44)- X(45)- X(46) - X(47)+ X(49)+ X(50)+2*X(51)+ X(52)+ X(53)- X(71)+2*X(54)- X(13)- X(57)- X(58) - X(12)+ X(62)+ X(63)- X(72) 
    X(28) = - X(64)+ X(5)+ X(68)+ X(26)+ X(31)+ X(33)+2*X(69)+ X(34)+ X(36)-3*K- X(40)- X(41) - X(15)+2*X(70)+ X(44)+ X(45)+ X(46)+ X(47)- X(49)- X(50)-2*X(51)- X(52)- X(53) + X(71)-2*X(54)+ X(13)+ X(57)+ X(58)+ X(12)- X(62)- X(63)+ X(72) 
    X(39) = - X(64)+ X(67)+2*X(68)+ X(7)+ X(26)+ X(3)+ X(4)+ X(29)+ X(31)+ X(32)+ X(33)+2*X(69)+ X(34) + X(35)+ X(36)- X(37)-2*X(38)- X(40)-2*X(41)- X(42)+2*X(70)-2*X(49)-2*X(50) -2*X(51)-2*X(52)-2*X(53)+2*X(71)-2*X(54)+ X(12)- X(63)+ X(72) 
    X(43) = X(23)- X(25)- X(26)+ X(3)- X(30)- X(31)- X(32)- X(33)- X(69)- X(34)- X(38)+ X(15)- X(70) - X(44)- X(45)- X(46)+ X(16)+ X(49)+ X(50)+ X(51)+ X(52)+ X(53)+ X(54)+ X(55)+ X(59) + X(60)+ X(63) 
    X(48) = X(64)- X(67)- X(23)-2*X(68)- X(7)+ X(25)-2*X(3)- X(4)- X(29)+ X(30)- X(69)- X(35)- X(36) +2*X(38)+6*K+ X(41)- X(15)- X(70)- X(47)- X(16)+ X(49)+ X(50)+ X(51)+ X(52) + X(53)-2*X(71)+ X(54)- X(55)- X(59)- X(60)- X(12)- X(72) 
    X(56) = 6*K- X(49)- X(50)- X(51)- X(52)- X(53)- X(54)- X(55)- X(57)- X(58)- X(59)- X(60) 
    X(6) = (-2*X(64)-2*X(66)+ X(67)+ X(23)+2*X(68)+4*X(7)+4*X(26)+2*X(3)+2*X(4)+4*X(29) -2*X(30)+4*X(31)+2*X(33)+2*X(69)+3*X(34)+2*X(35)+4*X(36)-4*X(38) -12*K-4*X(41)-2*X(42)+2*X(70)- X(45)+2*X(47)-3*X(49)- X(50)- X(51) - X(52)- X(53)+2*X(71)-5*X(54)+3*X(55)+2*X(13)+ X(57)+ X(58)+ X(59)+3*X(60) +3*X(12)-2*X(63)) /2 
    X(61) = - X(64)- X(65)- X(66)- X(67)- X(68)- X(69)+6*K- X(70)- X(71)- X(62)- X(63)- X(72) 
    X(8) = ( X(67)+ X(23)-2*X(30)- X(34)+2*X(37)-6*K+2*X(15)+ X(45)+ X(49)+ X(50)+ X(51) - X(52)+ X(53)+2*X(71)+ X(54)+ X(55)+ X(57)+ X(58)+ X(59)+ X(60)- X(12)) /2 
    X(9) = -(2*X(65)+ X(67)-2*X(5)- X(23)+2*X(68)+4*X(7)+4*X(26)+2*X(3)+2*X(4)+4*X(29) +2*X(31)+2*X(32)+2*X(33)+2*X(69)+ X(34)+2*X(35)+2*X(36)-2*X(37) -4*X(38)-6*K-4*X(41)-2*X(42)-2*X(44)- X(45)-2*X(46)-2*X(47) -2*X(16)-3*X(49)- X(50)+ X(51)- X(52)- X(53)+2*X(71)-3*X(54)+ X(55)- X(57) - X(58)- X(59)+ X(60)+3*X(12)+4*X(62)) /2

    I have made a ultra magic square of order 12 different positive integers not the general formula, and other means.

      19 1405  759  479  329  567 1219  193  885 1217 1049  699
    1069  663  129 1369  785  407  355  549 1317  169  899 1109
     959 1169 1077  613   79 1359  839  449  339  499 1279  159
    1249  179  869 1189 1029  729   49 1379  749  469  309  621
     259  559 1239  239  929 1179  979  691   39 1439  809  459
     841  429  369  529 1259  149  949 1149 1089  649   59 1349
     121 1411  821  381  321  521 1321  211  941 1101 1041  629
    1011  661   31 1431  779  491  291  541 1231  231  911 1211
     849 1161 1001  721   91 1421  741  441  281  601 1291  221
    1311  191  971 1131 1021  631  111 1391  857  393  301  511
     361  571 1301  153  921 1115 1063  685  101 1341  807  401
     771  421  253  585 1277  251  903 1141  991  711   65 1451

    K=1470, S=8820

    Now I check the general formula (see illustration).

    #351
    Natalia Makarova
    Partecipante

    I’m starting to find the ultra magic square of order 12 as follows

    I want to make a pattern of residues modulo 6, there is such a part of

    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0 
    5  1  1  5  1  5  5  5  1  5  1  1 
    5  5  1  1  5  1  1  5  5  1  1  5 
    1  5  5  1  1  5  5  1  5  5  1  1 
    5  5  1  5  1  1  1  5  1  5  5  1 
    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0 
    0  0  0  0  0  0  0  0  0  0  0  0

    Anybody can make a full pattern?

    #353
    Natalia Makarova
    Partecipante

    I made the pattern of residues modulo 6 for a ультра magic square of order 12.

    5 1 1 1 1 1 1 1 5 1 5 1
    1 5 1 1 1 1 5 1 1 1 1 5
    1 1 1 5 1 1 5 1 5 1 1 1
    1 1 5 5 5 1 5 1 1 1 5 5
    5 1 1 5 1 5 5 5 1 5 1 1
    5 5 1 1 5 1 1 5 5 1 1 5
    1 5 5 1 1 5 5 1 5 5 1 1
    5 5 1 5 1 1 1 5 1 5 5 1
    1 1 5 5 5 1 5 1 1 1 5 5
    5 5 5 1 5 1 5 5 1 5 5 5
    1 5 5 5 5 1 5 5 5 5 1 5
    5 1 5 1 5 5 5 5 5 5 5 1

    We have two sets of prime numbers in accordance with the residues modulo 6.

    Splitting the array into two groups according to the residues 1, 5.

    Group 1 (residue 1)

    19 31 37 43 61 97 103 109 151 163 181 193 211 241 277 283 307 367 373 379 409 421 439 457 487 499 523 541 607 613 631 643 661 673 709 727 751 
    769 787 811 823 829 853 877 883 907 967 991 1009 1021 1039 1051 1069 1087 1117 1123 1153 1201 1213 1231 1237 1279 1291 1297 1303 1321 1381 
    1399 1423 1429 1447 1453 1459

    Group 2 (residue 5)

    11 17 23 41 47 71 89 149 167 173 179 191 233 239 257 269 317 347 353 383 401 419 431 449 461 479 503 563 587 593 617 641 647 659 683 701 719 
    743 761 797 809 827 839 857 863 929 947 971 983 1013 1031 1049 1061 1091 1097 1103 1163 1187 1193 1229 1259 1277 1289 1307 1319 1361 1367 
    1373 1409 1427 1433 1439 1451
    #354
    Natalia Makarova
    Partecipante

    I found the approximation to a solution in which 16 errors:

     797   613  1201    43   109  1171  1117   511*  533*  787  1229   709 
    1069   149   103  1291   769   853   167  1453   895*  247*  907   917* 
    1021  1399   697* 1097* 1051   409   839    19   347   499  1279   163 
     661  1423   221*  461  1031   727   521  1381   751   211   317  1115* 
     257   607  1237   239   883  1193   983   659    37  1439   829   457 
     929   431   367   523  1277   151   877  1097  1091   643    61  1373 
      97  1409   827   379   373   593  1319   193   947  1103  1039   541 
    1013   641    31  1433   811   487   277   587  1231   233   863  1213 
     355* 1153  1259   719    89   949*  743   439  1009  1249    47   809 
    1307   191   971  1123  1451   631  1061   419   373*  773    71   449 
     553*  563  1223   575*   17  1303   617   701   179  1367  1321   401 
     761   241   683   937   959*  353   299* 1361  1427   269   857   673

    K=1470, S=8820

    Poor solution! It is difficult to find a good solution.

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