In Squareland, only squares live. Except for two exceptions, each of them has two friends, one with a perimeter $8 \mathrm{~cm}$ smaller and the other $8 \mathrm{~cm}$ larger. The average area of a Squareland square is $116 \mathrm{~cm}^{2}$. No two squares are the same, 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 length of a "Squarelander".

(Bednářová)
Let $a$ be the length of the side of the smallest of the squares. Then the lengths of the sides of the considered squares can be denoted as

$$
a, a+2, a+4, \ldots, a+2 t .
$$

Given the condition

$$
4 \cdot a=a+2 t
$$

we get

$$
3 a=2 t
$$

which means $a$ is even.

1. $a=2$. We calculate the average area (which must be $116 \mathrm{~cm}^{2}$ ).

$$
\frac{2^{2}+4^{2}+6^{2}+8^{2}}{4} \neq 116 \mathrm{~cm}^{2}
$$

2. $a=4$. We calculate the average area (which must be $116 \mathrm{~cm}^{2}$ ).

$$
\frac{4^{2}+6^{2}+8^{2}+10^{2}+12^{2}+14^{2}+16^{2}}{4}=116 \mathrm{~cm}^{2}
$$

3. $a=6$. We calculate the average area (which must be $116 \mathrm{~cm}^{2}$ ).

$$
\frac{6^{2}+8^{2}+\cdots+22^{2}+24^{2}}{4}=258 \mathrm{~cm}^{2}
$$

As $a$ increases, the average area also increases.

There are 7 inhabitants on the island. The side of the smallest one measures $4 \mathrm{~cm}$, the largest one $16 \mathrm{~cm}$, and the average side length is $10 \mathrm{~cm}$.