Question 974863: find a 3-by-3 magic square using the numbers 6,7,8,14,15,16,22,23 and 24
7 _ _
_ _ _
_ 8 _
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website!
Let the unknowns be A,B,C,D,E,F,G, like this
7 A B
C D E
F 8 G
I will number my equations in parentheses:
We sum all the numbers that go in the magic square:
(1) 6+7+8+14+15+16+22+23+24=135
therefore:
Each row, column, and diagonal must sum to 1/3 of 135 or 45
(2) 7+A+B = C+D+E = F+8+G = 7+C+F = A+D+8 = B+E+G = 7+D+G = B+D+F = 45
From (2) we get a system of 8 equations in 7 unknowns:
(3) A + B = 38
(4) C + D + E = 45
(5) F + G = 37
(6) C + F = 38
(7) A + D = 37
(8) B + E + G = 45
(9) D + G = 38
(10) B + D + F = 45
Eliminate A by subtracting (3)-(7), getting (11)
(4) C + D + E = 45
(5) F + G = 37
(6) C + F = 38
(8) B + E + G = 45
(9) D + G = 38
(10) B + D + F = 45
(11) B - D = 1
Eliminate C by subtracting (4)-(6) getting (12)
(5) F + G = 37
(8) B + E + G = 45
(9) D + G = 38
(10) B + D + F = 45
(11) B - D = 1
(12) D + E - F = 7
Eliminate E by subtracting (8)-(12) getting (13)
(5) F + G = 37
(9) D + G = 38
(10) B + D + F = 45
(11) B - D = 1
(13) B - D + F + G = 38
Eliminate D by adding (9)+(10)+(11)+(13) getting (14)
(5) F + G = 37
(14) 3B + 2F + 2G = 122
We eliminate F and G by multiplying (5) by -2 getting (15)
(15) - 2F - 2G = -74
and adding to (14)
(15) - 2F - 2G = -74
(14) 3B + 2F + 2G = 122
--------------------------------------
3B = 48
so B = 16
Substituting in (11) Substituting in (3)
(11) B - D = 1 A + B = 38
16 - D = 1 A + 16 = 38
-D = -15 A = 22
D = 15
Substituting in (9)
(9) D + G = 38
15 + G = 38
G = 23
Substituting in (5)
(5) F + G = 37
F + 23 = 37
F = 14
Substituting in (6)
(6) C + F = 38
C + 14 = 38
C = 24
Substituting C = 24 and D = 15 in (4)
(4) C + D + E = 45
24 + 15 + E = 45
39 + E = 45
E = 6
So the magic square
7 A B 7 22 16
C D E = 24 15 6
F 8 G 14 8 23
Edwin
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