A square wall has side length $n$ metres. Guillaume paints $n$ non-overlapping circular targets on the wall, each with radius 1 metre. Mathilde is going to throw a dart at the wall. Her aim is good enough to hit the wall at a single point, but poor enough that the dart will hit a random point on the wall. What is the largest possible value of $n$ so that the probability that Mathilde's dart hits a target is at least $\frac{1}{2}$ ?
The area of the square wall with side length $n$ metres is $n^{2}$ square metres.

The combined area of $n$ circles each with radius 1 metre is $n \cdot \pi \cdot 1^{2}$ square metres or $n \pi$ square metres.

Given that Mathilde hits the wall at a random point, the probability that she hits a target is the ratio of the combined areas of the targets to the area of the wall, or $\frac{n \pi \mathrm{m}^{2}}{n^{2} \mathrm{~m}^{2}}$, which equals $\frac{\pi}{n}$.

For $\frac{\pi}{n}>\frac{1}{2}$, it must be the case that $n<2 \pi \approx 6.28$.

The largest value of $n$ for which this is true is $n=6$.

ANSWER: $n=6$