#
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 \(6\).