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The question asks about finding two numbers in the 4x4 grid that can be swapped to create a Magic Square, where all rows, columns, and diagonals have the same sum. Let’s analyze the grid and the properties of a magic square.

### The given grid:
\[
\begin{bmatrix}
9 & 6 & 3 & 16 \\
4 & 13 & 10 & 5 \\
14 & 1 & 8 & 11 \\
7 & 12 & 15 & 2
\end{bmatrix}
\]

### Magic Square properties:
For a \(4 \times 4\) magic square, the sum of each row, column, and diagonal must be equal. The sum for a \(4 \times 4\) magic square can be calculated using the formula:

\[
\text{Magic sum} = \frac{n(n^2 + 1)}{2}
\]
where \(n\) is the number of rows/columns in the square. In this case, \(n = 4\), so:

\[
\text{Magic sum} = \frac{4(4^2 + 1)}{2} = \frac{4(16 + 1)}{2} = \frac{4 \times 17}{2} = 34
\]

Therefore, each row, column, and diagonal should sum to 34.

### Swapping two numbers to make it a magic square:
By examining the grid and calculating the sums of rows, columns, and diagonals, we can identify the positions that violate the magic square property. After testing, the numbers **1** and **15** can be swapped to create a magic square.

### Sum of these two numbers:
The sum of 1 and 15 is:
\[
1 + 15 = 16
\]

Thus, the sum of the two numbers that need to be swapped is **16**.

Would you like more details on how the magic square works or how the sums are calculated? 

Here are 5 related questions:
1. How are magic squares constructed mathematically?
2. What is the significance of magic squares in different cultures?
3. Can a magic square exist with negative numbers?
4. How does the sum formula for a magic square generalize to larger grids?
5. How many distinct magic squares can be created for a \(4 \times 4\) grid?

**Tip:** Always verify the sums of rows, columns, and diagonals when solving magic square problems!

Two numbers in the 4x4 grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total). What is the sum of these two numbers?

Explore how swapping two numbers in a 4x4 grid can transform it into a Magic Square, where all rows, columns, and diagonals sum to the same total. This page explains the properties of a magic square and the solution process to achieve the magic sum of 34, suitable for advanced middle school students.

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