Difference between revisions of "1999 CEMC Gauss (Grade 7) Problems/Problem 20"
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==Problem== | ==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. | 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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<math>\text{(A)}\ 32 \qquad \text{(B)}\ 28 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 16</math> | <math>\text{(A)}\ 32 \qquad \text{(B)}\ 28 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 16</math> | ||
==Solution== | ==Solution== | ||
Latest revision as of 12:21, 22 April 2025
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.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Solution
The sum of the first 9 positive odd integers is:
Thus, the sum of one of the columns, rows, or diagonals must be 
.
From the third column, we have:
= 
, and subtracting from both sides gives:
From the top row, we have:
, and subtracting 
from both sides gives:
From the diagonal containing A, C, and 3, we have:
, and subtracting 
from both sides gives:
The second column gives:
, and subtracting 
from both sides gives:
In the original problem, we were asked to find 
, which is equal to:
