The given problem presents a square grid where the sum of elements in each row, each column, and each of the two main diagonals is the same. This type of grid is known as a magic square. We need to find the values of the variables x, y, and z.

The square is represented as:

Magic Square Sum Calculation

Let S be the constant sum of each row, column, and diagonal.

We can find the sum S using the elements in the middle row or the main diagonal from top-left to bottom-right, as they involve the variable 'y'.

• Middle Row: $17 + y + 13 = S \implies 30 + y = S$
• Main Diagonal (top-left to bottom-right): $16 + y + 14 = S \implies 30 + y = S$

Both give us the same relationship: $S = 30 + y$.

Finding x, y, and z Values

Now, let's use other rows, columns, or the other diagonal to form equations involving x and z.

• Consider Row 1: $16 + 11 + x = S \implies 27 + x = S$
• Consider Column 3: $x + 13 + 14 = S \implies x + 27 = S$

From $27 + x = S$, substitute $S = 30 + y$:

$27 + x = 30 + y$

$x = 30 + y - 27$

$x = 3 + y$ (Equation 1)

• Consider Row 3: $z + 19 + 14 = S \implies z + 33 = S$
• Consider Column 1: $16 + 17 + z = S \implies 33 + z = S$

From $33 + z = S$, substitute $S = 30 + y$:

$33 + z = 30 + y$

$z = 30 + y - 33$

$z = y - 3$ (Equation 2)

Now let's use the other main diagonal (top-right to bottom-left) which involves x, y, and z:

• Main Diagonal (top-right to bottom-left): $x + y + z = S$

Substitute the expressions for x and z from Equation 1 and Equation 2 into this diagonal sum equation:

$(3 + y) + y + (y - 3) = S$

$3y = S$

Now we have two expressions for S: $S = 30 + y$ and $S = 3y$. We can equate them to solve for y:

$3y = 30 + y$

$3y - y = 30$

$2y = 30$

$y = \frac{30}{2}$

$y = 15$

Now that we have the value of y, we can find x and z using Equation 1 and Equation 2:

• Using Equation 1: $x = 3 + y = 3 + 15 = 18$
• Using Equation 2: $z = y - 3 = 15 - 3 = 12$

So, the values are x = 18, y = 15, and z = 12.

The question asks for the values corresponding to x, y and z, respectively, which are (18, 15, 12).

Verifying the Solution

Let's check if these values result in a magic square with a constant sum. The constant sum S should be $S = 30 + y = 30 + 15 = 45$. Let's fill the square with the calculated values:

Let's check the sums:

• Row 1: $16 + 11 + 18 = 45$
• Row 2: $17 + 15 + 13 = 45$
• Row 3: $12 + 19 + 14 = 45$
• Column 1: $16 + 17 + 12 = 45$
• Column 2: $11 + 15 + 19 = 45$
• Column 3: $18 + 13 + 14 = 45$
• Diagonal 1 (top-left to bottom-right): $16 + 15 + 14 = 45$
• Diagonal 2 (top-right to bottom-left): $18 + 15 + 12 = 45$

All sums are 45, confirming that our calculated values for x, y, and z are correct.

The values for x, y, and z are 18, 15, and 12 respectively, corresponding to (18, 15, 12).