On a $28.48 \mathrm{~cm}$ sized drawing paper, we drew a rectangle with sides at a distance of $d$ from each of the four sides, then constructed a square grid inside it, where the side of the smallest squares is also $d$. The total length of the drawn lines is 64.936 meters. How many squares with side $d$ are visible on the drawing and how many with sides $2 d, 3 d, \ldots, 28-2 d$?
The drawing sheet, including the borders, is divided into a strip of width $d$, with the number of strips being $28 / d$ or $48 / d$. The number of separating lines is one less than these, and their lengths are individually $48-2 d$ or $28-2 d \, \text{cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_3ac1cc5147d04d4fb8c8g-1.jpg?height=287&width=502&top_left_y=236&top_left_x=798)

Thus, the total length of the lines in cm is:

$$
\left(\frac{28}{d}-1\right)(48-2 d)+\left(\frac{48}{d}-1\right)(28-2 d)=6493.6
$$

From this, considering $d \neq 0$, the usual rearrangement leads to the equation $d^{2}-1680.4 d+672=0$, from which $d_{1}=0.4$, $d_{2}=1680$, the latter does not meet the requirements of the problem.

$$
\text{With } d=0.4, \text{ the grid contains } \left(\frac{28}{d}-2\right)\left(\frac{48}{d}-2\right)=68 \cdot 118=8024 \text{ smallest squares. The largest square in the grid }
$$

has a side length of $68 d$. If $k$ represents any of the numbers $1, 2, \ldots, 68$, we can mark a square with side length $k d$ at one corner of the grid. This leaves a segment of length $(118-k) d$ in the grid's length and $(68-k) d$ in its width, so it can be moved $118-k$ and $68-k$ times in the direction of the lines with steps of length $d$. Thus, each $k d$ side square is obtained exactly once, and the number of such squares is: $(119-k)(69-k)$

Maria Kiss (Bp. I., Building Materials Tech. IV. class)

Note. The number of rows and columns in the grid can be denoted by $m$ and $n$, respectively, and the questions can be answered without calculating $d$ by solving the system of equations

$$
(m+2) d=48, \quad(n+2) d=28, \quad[m(n+1)+n(m+1)] d=6493.6
$$

for $m$ and $n$ (since the problem does not ask for $d$).

Laborc Czékus (Bp. I. Toldy F. Gym. III. class)