4. On each field of the chessboard, a number is written. The sum of the numbers written on any four fields that form a knight's path (in the shape of the letter Г) is constant. How many different numbers are written on the board? Explain your answer.
Solution. Let's consider an arbitrary $3 \times 3$ section of the table (see diagram). According to the condition of the problem, it follows that $a_{1}+a_{4}+a_{5}+a_{6}=a_{4}+a_{5}+a_{6}+a_{3}$, from which it follows that $a_{1}=a_{3}$. Similarly, it can be shown that $a_{4}=a_{6}, a_{7}=a_{9}=a_{3}=a_{1}$, and $a_{2}=a_{8}$.

Then, from $a_{1}+a_{2}+a_{5}+a_{2}=a_{1}+a_{2}+a_{5}+a_{8}=a_{1}+a_{4}+a_{5}+a_{6}=a_{1}+a_{5}+a_{4}+a_{4}$

| $a_{1}$ | $a_{2}$ | $a_{3}$ |
| :--- | :--- | :--- |
| $a_{4}$ | $a_{5}$ | $a_{6}$ |
| $a_{7}$ | $a_{8}$ | $a_{9}$ |


| $a$ | $c$ | $a$ |
| :--- | :--- | :--- |
| $c$ | $a$ | $c$ |
| $a$ | $c$ | $a$ |

it follows that $a_{2}=a_{4}$. Similarly, it can be shown that $a_{1}=a_{3}=a_{5}=a_{7}=a_{9}=a$ and $a_{2}=a_{4}=a_{6}=a_{8}=c$. According to this, the numbers written on the same-colored cells (black or white) are equal. Let the number $a$ be written on the white cells, and the number $c$ be written on the black cells. If $a=c$, then the answer is one, and if $a \neq c$, then the answer is two.

## 2nd Year