Problem D

Throw to Win

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 \(n\)?