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 \(n\)?
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 \(6\).