Problem 3. In each of the 16 cells of a $4 \times 4$ square, one of the numbers $1,2,3, \ldots, 16$ is written. The sum of the numbers in each column is calculated. If one of the obtained sums is strictly greater than the other three, it is denoted by $S$.

a) Give an example of a completion of the square where $S=40$.

b) What is the smallest possible value of $S$?
Problem 3. In each of the 16 cells of a $4 \times 4$ square, one of the numbers $1,2,3, \ldots, 16$ is written. The sum of the numbers in each column is calculated. If one of the obtained sums is strictly greater than the other three, it is denoted by $S$.

a) Give an example of a completion of the square where $S=40$.

b) What is the smallest possible value of $S$?

## Solution

a) An example of such a completion is:

| 1 | 2 | 3 | $\mathbf{1 0}$ |
| :---: | :---: | :---: | :---: |
| 8 | 7 | 6 | $\mathbf{5}$ |
| 9 | 4 | 11 | $\mathbf{1 2}$ |
| 16 | 15 | 14 | $\mathbf{1 3}$ |

$\qquad$
b) The sum of the numbers written in the cells of the square is $1+2+3+\ldots+16=136$. Since $136=4 \cdot 34$, it follows that either the sum of the numbers in each column is 34, or there is a column whose sum is at least 35. In the first case, there is no $S$, and from the second case, it follows that $S$ is at least 35. To show that the minimum value of $S$ is 35, it remains to give an example of a completion of the square such that one column has a sum of 35, while the other columns have smaller sums. Here is such a completion:

| 1 | 2 | 3 | $\mathbf{4}$ |
| :---: | :---: | :---: | :---: |
| 8 | 7 | 6 | $\mathbf{5}$ |
| 9 | 10 | 11 | $\mathbf{1 2}$ |
| 16 | 15 | 13 | $\mathbf{1 4}$ |

(The sum of the numbers in the last column is 35, while the sums in the first three columns are 34, 34, and 33, respectively, so the last column is the one with $S=35$.)