We randomly throw a 10-forint coin onto a $40 \times 40$ cm chessboard. What is the probability that the coin

a) is completely inside one square?

b) partially covers exactly one side of a square?

c) partially covers two sides of a square, but not a corner?

d) covers a corner? (The diameter of the 10-forint coin is $19 \mathrm{~mm}$).
The "throwing onto the board" should be understood such that the center of the coin lands on the board. (Assuming that the entire extent of the coin lands within the board complicates the problem, as in this case the edge squares of the board play a special role.) Therefore, the center of the coin will certainly land inside or on the boundary of some square. Only this one square needs to be examined.

Considering the center, any point within the area of the square is possible. Since the side of the square is $\frac{40}{8}=5 \mathrm{~cm}$, the possible area is $50^{2}=2500 \mathrm{~mm}^{2}$.

a) In this case, the favorable area for the center of the coin is any point in the square whose distance from the nearest side of the square is $\geq$ the radius of the coin, i.e., $9.5 \mathrm{~mm}$. This area is a square whose side is $50-2 \cdot 9.5=31 \mathrm{~mm}$. (Marked as $a$ in the diagram.) Therefore, the desired probability is

$$
V_{a}=\frac{31^{2}}{50^{2}}=\frac{961}{2500}=0.3844
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_7e70f2c25623d227faf4g-1.jpg?height=390&width=382&top_left_y=691&top_left_x=861)

b) In this case, the favorable area for the center of the coin consists of 4 rectangles marked as $b$ in the diagram. The total area of these rectangles is $4 \cdot 31 \cdot 9.5=1178 \mathrm{~mm}$, and thus

$$
V_{b}=\frac{1178}{2500}=0.4712
$$

c) The favorable area for the center of the coin is the 4 regions marked as $c$ in the diagram. The total area of these regions is obtained by subtracting the area of the inscribed circle from the area of a square with side length $19 \mathrm{~mm}$, i.e., $19^{2}-9.5^{2} \pi=77.6 \mathrm{~mm}^{2}$, and thus

$$
V_{c} \approx \frac{77.6}{2500} \approx 0.0310
$$

d) The favorable area in this case is the 4 quarter-circles marked as $d$. The total area of these quarter-circles is $9.5^{2} \pi \approx 283.4 \mathrm{~mm}^{2}$, and thus

$$
V_{d} \approx \frac{283.4}{2500} \approx 0.1134
$$

László Bakó (Debrecen, Ref. g. II. o. t.)