Home › Forum › Ultra Magic Squares of prime numbers › Ultra magic square of 12-th order
- Questo topic ha 4 risposte, 1 partecipante ed è stato aggiornato l'ultima volta 8 anni, 10 mesi fa da Natalia Makarova.
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Maggio 17, 2015 alle 6:32 am #347Natalia MakarovaPartecipante
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 primes11 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
Perhaps make a ultra magic square of order 12 composed of the numbers of the array?
Maggio 19, 2015 alle 6:53 am #350Natalia MakarovaPartecipanteI 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).
Maggio 19, 2015 alle 7:48 pm #351Natalia MakarovaPartecipanteI’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?
- Questa risposta è stata modificata 8 anni, 10 mesi fa da Natalia Makarova.
Maggio 20, 2015 alle 10:58 am #353Natalia MakarovaPartecipanteI 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
Maggio 22, 2015 alle 8:49 pm #354Natalia MakarovaPartecipanteI 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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