On a $4 \times 4$ board, the numbers from 1 to 16 must be placed in the cells without repetition, such that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.

a) What is the Magic Sum of this board?
b) If the sum of the cells marked with $X$ in the board below is 34, what is the sum of the cells marked with $Y$?

| $Y$ |  |  | $Y$ |
| :--- | :--- | :--- | :--- |
|  | $X$ | $X$ |  |
|  | $X$ | $X$ |  |
| $Y$ |  |  | $Y$ |

c) If we fill in with consecutive natural numbers from $k$ to $(k+15)$, such that the Magic Sum is 50, what is the value of $k$?
Solution

a) Since there are 4 rows (as well as 4 columns), the Magic Sum is:

$$
\frac{1+2+3+\ldots+16}{4}=34
$$

b) If we add the two diagonals, we will have exactly the sum of the cells marked with $X$ and with $Y$. Thus, the sum of the cells marked with $Y$ is $2 \cdot 34-34=34$.

c) We have:

$$
\begin{aligned}
\frac{k+(k+1)+(k+2)+\ldots+(k+15)}{4} & =50 \\
16 k+(1+2+3+\ldots+15) & =200 \\
16 k+\frac{(1+15) 15}{2} & =200 \\
16 k+120 & =200 \\
16 k & =80 \\
k & =5
\end{aligned}
$$