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# help

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The following grid shows a magic square.  What is the sum of the three numbers in any row?

$$\begin{array}{c|c|c} 2x & 3 & 2 \\ \hline & & -3 \\ \hline 0 & x & \end{array}$$

Jun 7, 2020

#1
+21953
-1

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.

Jun 7, 2020
#2
+25541
+1

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:

$$\begin{array}{|rcll|} \hline \begin{array}{c|c|c} -2 & 3 & 2 \\ \hline 5 & 1& -3 \\ \hline 0 & -1 & 4 \end{array} \\ \hline \end{array} \begin{array}{rcll} 0+1+2 &=& 3 \\ -2+3+2 &=& 3 \\ 5+1-3 &=& 3 \\ 0-1+4 &=& 3 \\ -2+1+4 &=& 3 \\ -2+5+0 &=& 3 \\ 3+1-1 &=& 3 \\ 2-3+4 &=& 3 \\ \end{array}$$

Jun 8, 2020