Problem 1647

1. Define the Magic Square and its Properties:
A magic square of side $4$ is a $4 \times 4$ grid where the sum of the numbers in each row, column, and main diagonal is equal to $k$. Let the elements of the magic square be denoted as $a_{ij}$ where $i, j \in \{1, 2, 3, 4\}$.

2. Label the Corners:
Let the four corners of the magic square be labeled as follows:
$$\begin{array}{|c|c|c|c|} \hline a_{11} & a_{12} & a_{13} & a_{14} \\ \hline a_{21} & a_{22} & a_{23} & a_{24} \\ \hline a_{31} & a_{32} & a_{33} & a_{34} \\ \hline a_{41} & a_{42} & a_{43} & a_{44} \\ \hline \end{array}$$
The corners are $a_{11}, a_{14}, a_{41}, a_{44}$.

3. Sum of Rows and Columns:
Since the sum of each row and each column is $k$, we have:
$$a_{11} + a_{12} + a_{13} + a_{14} = k$$
$$a_{41} + a_{42} + a_{43} + a_{44} = k$$
$$a_{11} + a_{21} + a_{31} + a_{41} = k$$
$$a_{14} + a_{24} + a_{34} + a_{44} = k$$

4. Sum of Main Diagonals:
The sum of the main diagonals is also $k$:
$$a_{11} + a_{22} + a_{33} + a_{44} = k$$
$$a_{14} + a_{23} + a_{32} + a_{41} = k$$

5. Coloring the Grid:
Color the grid as follows:
$$\begin{array}{|c|c|c|c|} \hline A & X & X & A \\ \hline Y & B & B & Y \\ \hline Y & B & B & Y \\ \hline A & X & X & A \\ \hline \end{array}$$
Here, $A$ represents the corners, $X$ represents the other elements in the first and last rows and columns, and $B$ and $Y$ represent the elements in the middle rows and columns.

6. Equations from the Coloring:
From the coloring, we have:
$$A + X + X + A = 2A + 2X = k$$
$$Y + B + B + Y = 2Y + 2B = k$$
$$A + Y + Y + A = 2A + 2Y = k$$
$$X + B + B + X = 2X + 2B = k$$

7. **Solving for $A$, $B$, $X$, and $Y$:**
From the equations, we can see that:
$$2A + 2X = k \implies A + X = \frac{k}{2}$$
$$2Y + 2B = k \implies Y + B = \frac{k}{2}$$
$$2A + 2Y = k \implies A + Y = \frac{k}{2}$$
$$2X + 2B = k \implies X + B = \frac{k}{2}$$

8. **Equating $A$ and $B$:**
From the equations $A + X = \frac{k}{2}$ and $X + B = \frac{k}{2}$, we get:
$$A + X = X + B \implies A = B$$
Similarly, from $A + Y = \frac{k}{2}$ and $Y + B = \frac{k}{2}$, we get:
$$A + Y = Y + B \implies A = B$$

9. Sum of the Corners:
Since $A = B$, the sum of the numbers in the four corners is:
$$A + A + A + A = 4A$$
From $2A + 2X = k$, we have $A + X = \frac{k}{2}$. Since $A = B$, we can substitute $A$ for $X$ and get:
$$2A + 2A = k \implies 4A = k$$

Conclusion:
$$\boxed{k}$$