Problem 7 (Magic Square Problem)

Place three numbers a, b, and c in a 3 by 3 array as follows:

?	a	b
?	?	?
c	?	?

Find the numbers where the question marks are so the sum of the rows, colums, and two diagonals is always the same.
Solution
We label the 6 unknown entries u, v, w, z, y, and z as shown in the following table:

u	a	b
v	w	x
c	y	z

Our goal is to find equations these variables and then solve them in terms of the given numbers and . There is more than one way to do this, but we set

• Row 1 = Column 1: u + a + b = u + v + c

  Hence v = a = b - c

• Row 2 = Anti-Diagonal: v + w + x = c + w + b

  Hence, x = c + b - v = c + b - (a + b - c) = 2 c - a

• Row 2 = Column 2 : v + w + x = a + w + y

  Hence, y = (a + b - c) + (2c - a) - a = -a + b + c

• Diagonal = Row 3: u + w + z = c + y + z

  Hence, u + w = c + (-a + b + c) = -a + b + 2c

• Diagonal = Column 2: u + w + z = a + w + (-a + b + c)

  Hence u + z = b + c

• Diagonal = Column 1:u + w + z = u + v + c = u + (a + b - c) + c

  Hence, w + z = a + b

It is a simple matter to solve these last three equations, getting

u = (-2 a + b + 3 c)/2

w = (b + c)/2

z = (2 a + b - c)/2

Hence, we have

(-2 a + b + 3c)/2	(b + c)/2	b
(a + b - c)/2	(b + c)/2	2 c - a
c	-a + b + c	(2a + b - c)/2

If you check, you will find that the sum of every row, every column, and the two diagonals has the same sum of

common sum = (3b + 3c )/2

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Last modified on Friday, February 12, 1999