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1999 CEMC Gauss (Grade 7) Problems/Problem 20

Problem

The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of A + E.

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$\text{(A)}\ 32 \qquad \text{(B)}\ 28 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 16$

Solution

The sum of the first 9 positive odd integers is:

$1 + 3 + 5 + ... + 17 = 9^2 = 81$

Thus, the sum of one of the columns, rows, or diagonals must be $\frac{81}{3} = 27$.

From the third column, we have:

$B + 13 + 3 = 27$

$13 + 3$ = $16$, and subtracting $16$ from both sides gives:

$B = 11$

From the top row, we have:

$A + 1 + B = A + 1 + 11 = 27$

$1 + 11 = 12$, and subtracting $12$ from both sides gives:

$A = 15$

From the diagonal containing A, C, and 3, we have:

$C + A + 3 = C + 15 + 3 = 27$

$15 + 3 = 18$, and subtracting $18$ from both sides gives:

$C = 9$

The second column gives:

$E + 1 + C = E + 1 + 9 = 27$

$1 + 9 = 10$, and subtracting $10$ from both sides gives:

$E = 17$

In the original problem, we were asked to find $A + E$, which is equal to:

$A + E = 15 + 17 = \boxed {\textbf {(A)} 32}$

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