Question 774240: solve these magic squares, all rows,columns and diagonals all line up to the same number:
-6 8 -2 ? ? 5 ? ? ?
? ? ? ? -12 ? ? -12 -16
? -8 ? ? -8 3 -10 ? ?
6 ? 4 -18
? ? -6 ?
? ? -8 10
12 ? ? -12
Answer by Edwin McCravy(20060) (Show Source):
You can put this solution on YOUR website!
I'll do only the last one, the hardest one. The others
are similar but easier. However there is no solution to
the second one. Put different letters in place of the
question marks:
A B 5
C -12 D
E -8 3
The diagonal sum 5-12+E (I'll label that Diag2) cannot equal to the
bottom row sum (I'll label that Row3)
5-12+E=E-8+3
-7+E=E-5
-7=-5
A contradiction. So a solution is impossible.
Did you perchance type a number wrong? (The others have solutions).
--------------------------------------------
Here's the last one, the hardest one:
Put different letters in place
of all the question marks:
6 A 4 -18
B C -6 D
E F -8 10
12 G H -12
Write out the row, column, and diagonal sums
and simplify them:
Row1: 6+A+4-18 = A-8
Row2: B+C-6+D = B+C+D-6
Row3: E+F-8+10 = E+F+2
Row4: 12+G+H-12 = G+H
Col1: 6+B+E+12 = B+E+18
Col2: A+C+F+G = A+C+F+G
Col3: 4-6-8+H = H-10
Col4: -18+D+10-12 = D-20
Diag1:0 6+C-8-12 = C-14
Diag2: -18-6+F+12 = F-12
Row3 and Diag2 contain only E and F, so we set their sums equal:
E+F+2 = F-12
Subtract F from both sides:
E = -14
Row4 and Col3 contain only G and H, so we set their sums equal:
G+H = H-10
Subtract H from both sides
G = -10
So we substitute those in, and our list of sums get
a couple more columns on the right
Row 1: 6+A+4-18 = A-8
Row 2: B+C-6+D = B+C+D-6
Row 3: E+F-8+10 = E+F+2 = -14+F+2 = F-12
Row 4: 12+G+H-12 = G+H = -10+H = H-10
Col 1: 6+B+E+12 = B+E+18 = B-14+18 = B+4
Col 2: A+C+F+G = A+C+F+G = A+C+F-10
Col 3: 4-6-8+H = H-10
Col 4: -18+D+10-12 = D-20
Diag1: 6+C-8-12 = C-14
Diag2: -18-6+F+12 = F-12
Now let's get things in terms of A:
A-8=F-12 A-8=H-10 A-8=B+4 A-8=D-20 A-8=C-14
A+4=F A+2=H A-12=B A+12=D A+6=C
Set Row1 = Row2, simplify, then substitute the above in the
expression for Row2:
B+C+D-6=A-8
B+C+D+2=A
(A-12)+(A+6)+(A+12)+2=A
A-12+A+6+A+12+2=A
3A+8=A
2A=-8
A=-4
So
A+4=F A+2=H A-12=B A+12=D A+6=C
-4+4=F -4+2=H -4-12=B -4+12=D -4+6=C
0=F -2=H -16=B 8=D 2=C
And we had found the values E=-14 and G=-10 first.
That's it.
6 A 4 -18 6 -4 4 -18
B C -6 D -16 2 -6 8
E F -8 10 -14 0 -8 10
12 G H -12 12 -10 -2 -12
You do the others.
Edwin
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