Ultra magic square of 8-th order

[Natalia Makarova]

Scheme for ultra magic square of order 8

The general formula of ultra magic square of order 8

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

We have 18 free variables and 14 dependent variables, if the constant assotsiativity K is set.
I used the general formula and following pattern of residues modulo 3

1 1 1 2 2 1 2 2 
1 2 1 2 1 2 2 1 
1 2 2 2 2 1 1 1 
2 2 1 1 2 1 2 1 
2 1 2 1 2 2 1 1 
2 2 2 1 1 1 1 2 
2 1 1 2 1 2 1 2 
1 1 2 1 1 2 2 2

I found a minimal solution with magic constant S = 2040. This solution is presented for the contest. You’ll see it after the contest.

[Autore]

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