Fill In Each Blank With A Number Or Expression Such That Each Row And Column Has The Same Sum. \[ \begin{tabular}{|c|c|c|} \hline \square$ & X + 2 X+2 X + 2 & 2 − X 2-x 2 − X \ \hline 5 − X 5-x 5 − X & X X X & □ \square □ \ \hline □ \square □ & 2 & □ \square □

Introduction

A magic square is a square grid filled with distinct positive integers in the range 1 to n^2, such that each row, column, and diagonal sums up to the same constant. In this article, we will explore a specific magic square puzzle where we need to fill in the blanks with a number or expression such that each row and column has the same sum. We will use mathematical reasoning and problem-solving skills to find the solution.

The Magic Square Puzzle

The given magic square puzzle is as follows:

	$x+2$	$2-x$
$5-x$	$x$	$\square$
$\square$	2	$\square$

Our goal is to fill in the blanks with a number or expression such that each row and column has the same sum.

Analyzing the First Row

Let's start by analyzing the first row. We know that the sum of the first row is equal to the sum of the other rows. Therefore, we can write an equation based on the first row:

$(x+2) + (2-x) + \square = \text{sum}$

Simplifying the equation, we get:

$4 + \square = \text{sum}$

Analyzing the Second Row

Now, let's analyze the second row. We know that the sum of the second row is equal to the sum of the other rows. Therefore, we can write an equation based on the second row:

$(5-x) + x + \square = \text{sum}$

Simplifying the equation, we get:

$5 + \square = \text{sum}$

Analyzing the Third Row

Finally, let's analyze the third row. We know that the sum of the third row is equal to the sum of the other rows. Therefore, we can write an equation based on the third row:

$\square + 2 + \square = \text{sum}$

Simplifying the equation, we get:

$2 + 2\square = \text{sum}$

Finding the Sum

Since each row and column has the same sum, we can set up an equation based on the sum of the first row:

$4 + \square = \text{sum}$

We can also set up an equation based on the sum of the second row:

$5 + \square = \text{sum}$

Equating the two equations, we get:

$4 + \square = 5 + \square$

Simplifying the equation, we get:

$0 = 1$

This is a contradiction, which means that our initial assumption is incorrect. Therefore, we need to re-examine our equations.

Re-examining the Equations

Let's re-examine the equations we derived earlier:

$4 + \square = \text{sum}$

$5 + \square = \text{sum}$

$2 + 2\square = \text{sum}$

We can see that the first two equations are identical, which means that we have two equations with two unknowns. We can solve this system of equations using substitution or elimination.

Solving the System of Equations

Let's use substitution to solve the system of equations. We can solve the first equation for $\square$:

$\square = \text{sum} - 4$

Substituting this expression into the second equation, we get:

$5 + (\text{sum} - 4) = \text{sum}$

Simplifying the equation, we get:

$\text{sum} - 4 + 5 = \text{sum}$

This equation is true for any value of $\text{sum}$, which means that we have an infinite number of solutions.

Finding the Value of $\square$

Since we have an infinite number of solutions, we can choose any value for $\text{sum}$. Let's choose a value of 15 for $\text{sum}$. Substituting this value into the equation, we get:

$4 + \square = 15$

Solving for $\square$, we get:

$\square = 11$

Finding the Values of $x$ and $2-x$

Now that we have found the value of $\square$, we can find the values of $x$ and $2-x$. We can substitute the value of $\square$ into the first row:

$(x+2) + (2-x) + 11 = 15$

Simplifying the equation, we get:

$x + 2 + 2 - x + 11 = 15$

This equation is true for any value of $x$, which means that we have an infinite number of solutions.

Finding the Value of $x$

Since we have an infinite number of solutions, we can choose any value for $x$. Let's choose a value of 3 for $x$. Substituting this value into the equation, we get:

$(3+2) + (2-3) + 11 = 15$

Simplifying the equation, we get:

$5 - 1 + 11 = 15$

This equation is true, which means that our choice of $x$ is correct.

Conclusion

In this article, we solved a magic square puzzle where we needed to fill in the blanks with a number or expression such that each row and column has the same sum. We used mathematical reasoning and problem-solving skills to find the solution. We found that the value of $\square$ is 11, and the value of $x$ is 3. We also found that the value of $2-x$ is -1.