Maximum Number Of Darts Inside Of A Circular Dartboard - Leetcode Solution

Problem statement

You have a circular dartboard of radius r and an infinite number of darts. Each dart can be represented by a point in the Cartesian plane, and the distance from the origin of the plane to the dart is called its Euclidean distance. You need to throw the darts to the board in such a way that they are all inside it, and you want to maximize the number of darts inside the board. A dart is considered to be inside if it lies on or within the circle.

Write a function maxPoints() that takes an integer r as input and returns the maximum number of darts that can be placed inside the dart board with radius r. The answer should be accurate to two decimal places.

Example:

Input: r = 2
Output: 4.00
Explanation: 
You can place four darts inside the circle with radius 2, such that the distance between every pair of them is at least 2.

Solution

The main idea behind the problem is that we need to place as many darts as possible inside the circular dartboard. Since all the darts are represented by a point in the Cartesian plane, they can be arranged in a two-dimensional grid. The optimal way to arrange the darts inside the board is to place them in a regular polygonal arrangement.

The regular polygonal arrangement is formed by dividing the circle into equal segments and placing the darts at the endpoints of these segments. The number of segments needed to form a regular polygon with n vertices is given by the function:

n = ceil(2 * pi * r / (2 * r))

where ceil(x) is the smallest integer greater than or equal to x. Once we know the number of segments, we can place the darts as follows:

• Let theta be the central angle of the polygon (i.e., the angle formed by any two adjacent points). We can compute this angle as:

theta = 2 * pi / n

• We can place the first dart at the origin of the plane, and then locate the remaining n-1 darts as follows:

  • For each i from 1 to n-1, compute the polar coordinates of the ith endpoint as (r, i * theta).
  • Convert the polar coordinates to Cartesian coordinates using the conversion formula x = r * cos(theta), y = r * sin(theta).
  • Place the ith dart at the point (x, y).

Finally, we can compute the maximum number of darts that can be placed inside the dartboard by dividing the area of the circle by the area of a single dart. The area of a dart is given by:

A = pi * (r / n)^2

Therefore, the maximum number of darts is given by:

max_darts = floor(pi * r^2 / A)

where floor(x) is the largest integer less than or equal to x.

Implementation

Here's the Python implementation of the maxPoints() function:

import math

def maxPoints(r: int) -> float:
    n = math.ceil(2 * math.pi * r / (2 * r))
    theta = 2 * math.pi / n
    area = math.pi * (r / n) ** 2
    max_darts = math.floor(math.pi * r ** 2 / area)
    return round(max_darts, 2)

The function takes an integer r as input, computes the number of segments n, the central angle theta, and the area of a single dart. Then, it computes the maximum number of darts that can be placed inside the dartboard and returns the result rounded to two decimal places.

Time complexity: O(1)

Space complexity: O(1)

Conclusion

In this article, we have solved the Maximum Number Of Darts Inside Of A Circular Dartboard problem on LeetCode. We have presented a simple and efficient algorithm that computes the maximum number of darts that can be placed inside the dartboard with radius r. The algorithm uses a regular polygonal arrangement of the darts and computes the maximum number of darts using simple geometric formulas. The time and space complexity of the algorithm are both constant, making it suitable for large inputs.

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