 # Problem of the Week Problem D and Solution Throw to Win

## Problem

Kurtis is creating a game for a math fair. They attach $$n$$ circles, each with radius $$1$$ metre, onto a square wall with side length $$n$$ metres, where $$n$$ is a positive integer, so that none of the circles overlap. Participants will throw a dart at the wall and if the dart lands on a circle, they win a prize. Kurtis wants the probability of winning the game to be at least $$\frac{1}{2}$$.

If they assume that each dart hits the wall at a single random point, then what is the largest possible value of $$n$$? ## Solution

The area of the square wall with side length $$n$$ metres is $$n^2$$ square metres.

The area of each circle is $$\pi (1)^2=\pi$$ square metres. Since there are $$n$$ circles, the total area covered by circles is $$n\pi$$ square metres.

If each dart hits the wall at a single random point, then the probability that a dart lands on a circle is equal to the area of the wall covered by circles divided by the total area of the wall. That is, $\dfrac{n\pi~\text{square metres}}{n^2~\text{square metres}}=\dfrac{\pi}{n}$

If this probability must be at least $$\frac{1}{2}$$, then \begin{aligned} \frac{\pi}{n}&\ge \frac{1}{2}\\ \pi &\ge \dfrac{n}{2},\quad \text{since }n>0\\ 2\pi &\ge n\\ n & \le 2\pi \approx 6.28\ \end{aligned} Thus, since $$n$$ is an integer, the largest possible value of $$n$$ is $$6$$.