Ultra magic square 7th order

[Natalia Makarova]

The scheme of ultra magic square of order 7:

X(1)  X(2)  X(3)  X(4)  X(5)  X(6)  X(7) 
X(8)  X(9)  X(10)  X(11)  X(12)  X(13)  X(14) 
X(15)  X(16)  X(17)  X(18)  X(19)  X(20)  X(21) 
X(22)  X(23)  X(24)  K/2  K-X(24)  K-X(23)  K-X(22) 
K-X(21)  K-X(20)  K-X(19)  K-X(18)  K-X(17)  K-X(16)  K-X(15) 
K-X(14)  K-X(13)  K-X(12)  K-X(11)  K-X(10)  K-X(9)  K-X(8) 
K-X(7)  K-X(6)  K-X(5)  K-X(4)  K-X(3)  K-X(2)  K-X(1)

There K – associative constant square, S = 7 * K / 2, if S – magic constant square.

The general formula of ultra magic square of order 7:

X(1) = (2*X(18)-2*X(20)- K+2*X(12)+2*X(6)-2*X(14)+2*X(24))/2 
X(10) = - X(16)+ X(3)+ X(18)+ X(5)- X(20)- X(21)- K+ X(12)+2*X(6)-2*X(14)+ X(15)+ X(24) 
X(11) = -(-2*X(16)+4*X(3)-2*X(20)-2*X(21)+3*K-2*X(23)-4*X(14)-2*X(15)+2*X(24))/2 
X(13) = X(16)- X(18)- X(5)+3*K- X(12)-2*X(6)- X(15) 
X(17) = -(2*X(3)+2*X(18)-5*K+2*X(6)-2*X(14)+2*X(15)+2*X(24))/2 
X(19) = - X(16)+ X(3)- X(20)- X(21)+ K+ X(6)- X(14)+ X(24) 
X(2) = -(-2*X(16)+4*X(3)+2*X(18)-2*X(20)- K-2*X(23)+2*X(6)-4*X(14)+2*X(24))/2 
X(22) = - X(16)+ X(3)- X(5)+ X(20)+ K- X(23) 
X(4) = -(-2*X(3)+2*X(18)+2*X(5)-2*X(21)-7*K+2*X(23)+4*X(12)+4*X(6)+2*X(15))/2 
X(7) = - X(16)+ X(18)- X(21)+ X(12)+ X(6)- X(14)+ X(15) 
X(8) = - X(3)+ X(5)+ X(23)+ X(14)- X(24) 
X(9) = - X(16)+2*X(3)- X(5)+3*K-2*X(23)- X(12)-2*X(14)- X(15)+ X(24)

For a given associative constant square K have 12 free variables and 12 dependent variables.

I am now trying to find a minimal solution with magic constant S = 4487.
The array of primes for this solution:

5 23 53 59 89 101 131 173 179 191 233 251 263 269 311 353 401 419 443 461 509 521 563 599 641 683 719 761 773 821 839 863 881 929 971 1013 1019 1031 1049 1091 1103 1109 1151 1181 1193 1223 1229 1259 1277

I found some solutions with one error, for example:

509  131  1031 1049 1193 401  173 
1091 821  839  53   863  761  59 
263  599  695* 563  179  929  1259 
269  1181 311  641  971  101  1013 
23   353  1103 719  587  683  1019 
1223 521  419  1229 443  461  191 
1109 881  89   233  251  1151 773

K=1282, S=4487

[Natalia Makarova]

I have proved that the solution with the magic constant S = 4613 for n = 7 is minimal.

[Autore]

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