## Task 3 - 130913

The numbers 1, 2, 3, and 4 are to be entered into the adjacent square such that in each row, each column, and each of the two diagonals, each of the four numbers appears exactly once. Three places are already filled with numbers and should remain unchanged.

Investigate whether such an entry is possible and whether only one entry possibility exists. If this is the case, perform the entry.

Hint: For describing the solution, use the letters entered at the edges of the square.

Example: The number 2 is already entered in field bC.

| $\mathrm{a}$ | 1 |  |  |  |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{b}$ |  |  | 2 |  |
| $\mathrm{c}$ |  |  |  | 3 |
| $\mathrm{d}$ |  |  |  |  |
|  | $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |
For the number 2, only the field $B a$ is possible in row $a$, since $C a$ is ruled out by the column and $D a$ by the diagonal. Then the number 3 must go into the box $C a$, because it cannot be in $B a=2$ and $D a$ (column). The 4 then goes into $D a$. This looks like this:

It follows (uniquely) that $D d=2$ ($D d$ cannot be 1 due to the diagonal and 3 or 4 due to the column). $D b$ becomes 1 (column). Furthermore, it follows that $A d=3 (A d \neq 1$ due to the column, $A d \neq 2$ due to the row, $A d \neq 4$ due to the diagonal) and $B c=1$ (diagonal). This then looks like this:

Furthermore, $A b=4 (A b \neq 1, A b \neq 3$ due to the column; $A b \neq 2$ due to the row) and $A c=2$ (column) must hold. Accordingly, $B b=3$ (row) and $B d=4$ (column) follow. Then $C c=4$ (row) and $C d=1$ (column) also follow. The complete and unique solution then looks like this:

| $a$ | 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: | :---: |
| $b$ | 0 | 0 | 2 | 0 |
| $c$ | 0 | 0 | 0 | 3 |
| $d$ | 0 | 0 | 0 | 0 |
|  | $A$ | $B$ | $C$ | $D$ |
| $a$ | 1 | 2 | 3 | 4 |
| $b$ | 0 | 0 | 2 | 1 |
| $c$ | 0 | 1 | 0 | 3 |
| $d$ | 3 | 0 | 0 | 2 |
|  | $A$ | $B$ | $C$ | $D$ |
| $a$ | 1 | 2 | 3 | 4 |
| $b$ | 4 | 3 | 2 | 1 |
| $c$ | 2 | 1 | 4 | 3 |
| $d$ | 3 | 4 | 1 | 2 |
|  | $A$ | $B$ | $C$ | $D$ |

Incidentally, there are a total of 48 possibilities to construct a $4 \times 4$ square with the mentioned properties (without the pre-assignment).