In Squareland, only squares live. Except for two exceptions, each of them has two friends, one of whom has a perimeter that is $8 \, \text{cm}$ smaller and the other $8 \, \text{cm}$ larger. The average side length of a Squareland square is $15 \, \text{cm}$. No two squares are identical, and the perimeter of the smallest square is equal to the side length of the largest square. Determine:

a) The number of inhabitants in Squareland,

b) the dimensions of the largest and smallest Squareland square,

c) the average area of a "Squarelandian".

(Bednářová)
Given that the perimeters of the squares differ by $8 \mathrm{~cm}$, their sides must differ by $2 \mathrm{~cm}$.

Let's denote the sides of the sought squares as $a, a+2, a+4, \ldots, a+2 t$. From the problem statement, it is clear that the average side length of the largest and smallest square is $15 \mathrm{~cm}$. Therefore,

$$
\frac{a+(a+2 t)}{2}=15
$$

Furthermore, it holds that

$$
4 a=a+2 t .
$$

By solving this system of equations, we get $a=6, t=9$. All the sides of the squares are: $6,8,10$, $12,14,16,18,20,22$ and 24. Now we can easily answer the questions a), b), c).
a) There are 10 inhabitants in Squareland.

b) The smallest Squareland square has dimensions $6 \times 6$ and the largest is $24 \times 24$.

c) The average area of a Squareland square is

$$
\frac{6^{2}+8^{2}+\ldots+22^{2}+24^{2}}{10}=258 \mathrm{~cm}^{2} .
$$

(2 points for each of the three parts)