The following grid shows a magic square. What is the sum of the three numbers in any row?
\(\begin{array}{c|c|c} 2x & 3 & 2 \\ \hline & & -3 \\ \hline 0 & x & \end{array}\)
Place a "?" into the middle square.
Adding down the middle column: 3 + ? + x
Adding the diagonal from lower-left to upper-right: 0 + ? + 2
Since these sums must be the same: 3 + ? + x = 0 + ? + 2
Subtracting the ?: 3 + x = 0 + 2
Solving: x = -1
Since x = -2, the value of the upper-left-hand box = 2x = 2(-1) = -2.
So, the top row has a sum of: -2 + 3 + 2 = 3
which is the sum of every row, every column, and every diagonal.
It is now possible to find the value of each box.
The following grid shows a magic square.
What is the sum of the three numbers in any row?
\(\begin{array}{c|c|c} 2x & 3 & 2 \\ \hline & & -3 \\ \hline 0 & x & \end{array}\)
Find x:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} 2x & 3 & 2 \\ \hline & & -3 \\ \hline 0 & x & \color{red}y \end{array} \\ 0+x+y &=& 2-3+y \\ x &=& 2-3 \\ \mathbf{x}&=& \mathbf{-1} \\ \hline \end{array}\)
The sum of the three numbers in any row:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} \color{red}-2 & \color{red}3 & \color{red}2 \\ \hline & & -3 \\ \hline 0 & -1 & \end{array} \\ -2+3+2 &=& \mathbf{3} \\ \hline \end{array}\)
Find y:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} -2 & 3 & 2 \\ \hline & \color{red}y& -3 \\ \hline 0 & -1 & \end{array} \\ 0+y+2 &=& 3 \\ y &=& 3-2 \\ \mathbf{y}&=& \mathbf{1} \\ \hline \end{array}\)
Find y:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} -2 & 3 & 2 \\ \hline & 1& -3 \\ \hline 0 & -1 & \color{red}y \end{array} \\ -2+1+y &=& 3 \\ -1+y &=& 3\\ y &=& 3+1\\ \mathbf{y}&=& \mathbf{4} \\ \hline \end{array}\)
Find y:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} -2 & 3 & 2 \\ \hline\color{red}y & 1& -3 \\ \hline 0 & -1 & 4 \end{array} \\ -2+y+0 &=& 3 \\ y &=& 3+2 \\ \mathbf{y}&=& \mathbf{5} \\ \hline \end{array}\)
The magic square:
\(\begin{array}{|rcll|} \hline \begin{array}{c|c|c} -2 & 3 & 2 \\ \hline 5 & 1& -3 \\ \hline 0 & -1 & 4 \end{array} \\ \hline \end{array} \begin{array}{rcll} 0+1+2 &=& 3 \\ -2+3+2 &=& 3 \\ 5+1-3 &=& 3 \\ 0-1+4 &=& 3 \\ -2+1+4 &=& 3 \\ -2+5+0 &=& 3 \\ 3+1-1 &=& 3 \\ 2-3+4 &=& 3 \\ \end{array}\)