3. In the $3 \times 3$ grid shown, each of the three symbols has a different value. The sum of the values of the symbols in each row is given to the right of that row, and the sum of the values of the symbols in each column is given below that column. What is the value of $x$ ?
3. Solution 1
Each of the three column sums represents the total of the three entries in that column. Therefore, the sum of the three column sums is equal to the sum of the nine entries in the table.
Thus, the sum of the nine entries in the table is $22+12+20=54$.
Similarly, the sum of the three row sums must equal the sum of the nine entries in the table. Thus, $x+20+15=54$ or $x=19$.
Solution 2
From the second column, $\triangle+\triangle+\triangle=12$ or $3 \times \triangle=12$ and so $\triangle=4$.
From the second row, $\square+\triangle+\square=20$.
Since $\triangle=4$, then $2 \times \square=16$ and so $\square=8$.
From the first column, $\varnothing+\square+\varnothing=22$.
Since $\square=8$, then $2 \times \varnothing=14$ and so $\odot=7$.
From the first row, $x=\Omega+\triangle+\square=7+4+8=19$.
ANSWER: $x=19$