Magic Square Algebraic Sum

Question

Rahul has just made a $3 	imes 3$ magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The entries in the cells are given as expressions in x, y, and Z: | $3x+4y$ | $2x$ | $2x+y+z$ | | $2x^2$ | $4y$ | $y^2+z$ | | $y+z$ | $3x+2z$ | $z-1$ | Find N?

Accepted Answer

B. Equating sums of rows and columns: Sum of row 3 = $y+z + 3x+2z + z-1 = 3x+y+4z-1$. Sum of col 2 = $2x + 4y + 3x+2z = 5x+4y+2z$. Equating diagonal 1 sum: $(3x+4y) + 4y + (z-1) = 3x+8y+z-1$. Solving these simultaneous equations gives $x = 2$, $y = 3$, $z = 7$. Substitute $x=2, y=3, z=7$ into row 1: $(3(2)+4(3)) + 2(2) + (2(2)+3+7) = 18 + 4 + 14 = 36$. Hence, $N = 36$.

Answer

12

Answer

36

Answer

21

Answer

40

Answer

24

Magic Square Algebraic Sum

Explanation

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