We have squares $A B C D$ and $K L M N$. The side lengths of both squares are expressed in centimeters as whole numbers. Point $K$ is an internal point on segment $A B$, point $L$ lies at point $B$, and point $M$ is an internal point on segment $B C$. The area of the hexagon $A K N M C D$ is $225 \mathrm{~cm}^{2}$. What could be the perimeter of this hexagon? Find all possibilities.

(L. Šimünek)
The length of the side of square $A B C D$ is denoted by $a$ and the length of the side of square $K L M N$ is denoted by $b$, both in centimeters (see the figure). The problem asks for the perimeter of the hexagon $A K N M C D$, which is the same as the perimeter of square $A B C D$, i.e., $4 a$.

![](https://cdn.mathpix.com/cropped/2024_04_17_3afc6eb4999c6025c30cg-3.jpg?height=594&width=602&top_left_y=1693&top_left_x=727)

The area of the hexagon $A K N M C D$ is given by

$$
a^{2}-b^{2}=(a-b) \cdot(a+b)=225 .
$$

Since, according to the problem, point $K$ is an internal point of segment $A B$ and $L=B$, it is clear that $b \neq 0$ and $b>0$ and $c>0$. The lengths $a, b$ are integers according to the problem, and the length $c$ is also an integer because $c=a-b$.

The area of the hexagon $A K N M C D$ can be expressed as:

$$
225=2 b c+c^{2}
$$

We will substitute natural numbers for $c$ in this equation and determine each time whether $b$ is also a natural number. The following table shows this process, omitting the verification of all even $c$. If $c$ were even, the expression $2 b c$ would be odd according to the above equation, and $b$ would not be an integer.

| $c$ | $2 b c=225-c^{2}$ | $b=2 b c: 2 c$ |
| :---: | :---: | :---: |
| 1 | 224 | 112 |
| 3 | 216 | 36 |
| 5 | 200 | 20 |
| 7 | 176 | 176 is not divisible by 7 |
| 9 | 144 | 8 |
| 11 | 104 | 104 is not divisible by 11 |
| 13 | 56 | 56 is not divisible by 13 |
| 15 | 0 | 0 |

We have already stated that $b>0$, so the last row does not lead to a solution. We do not continue searching for possible solutions because $b$ would clearly be negative. Therefore, the problem has exactly four solutions. The perimeter of the hexagon $A K N M C D$ can be calculated as $4 b+4 c$. The perimeter of the given hexagon can thus be $68 \mathrm{~cm}, 100 \mathrm{~cm}, 156 \mathrm{~cm}$, or $452 \mathrm{~cm}$.

Evaluation. 1 point for each resulting perimeter; 2 points for the explained method, which must show that there are no more solutions.