1998 CEMC Gauss (Grade 7) Problems/Problem 15

Problem

The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of n?

[A 3x3 magic square grid is shown. 8 is in the 1st row 1st column. 9 is in the 2nd row 1st column. 4 is in the 2nd row 3rd column. 4 is in the 3rd row 1st column. $n$ is in the 3rd row 2nd column.]

$\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

Solution

The "magic" sum is $8 + 9 + 4 = 21,$ so the center square (2nd row 2nd column) is $21 - 9 - 5 = 7.$

The square in the lower right (3rd row 3rd column) has value 6, therefore $4 + n + 6 = 21.$

The answer is $\text{(E)} \quad 11.$

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