Maximum Number of Darts Inside of a Circular Dartboard

This problem is about finding the optimal position for a dartboard of radius r so that the maximum number of darts can lie on the dartboard. A dart will lie on the dartboard if the distance from the dart to the center of the dartboard is less than or equal to r.

Here’s a step-by-step solution:

  1. Iterate through each pair of darts: For every pair of darts, you can find two possible centers for the dartboard that is at a distance r from both darts. To find these centers, you need to solve the equations for two circles with the same radius but different centers, where the centers are the positions of the two darts.

  2. Check the darts for each possible center: For each possible center, you can iterate through all the darts and check if the distance from the dart to the center is less than or equal to r. If so, this dart will lie on the dartboard.

  3. Find the maximum number of darts: Track the maximum number of darts that can lie on the dartboard for each possible center.

  4. Return the result: Return the maximum number of darts.

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from math import sqrt

class Solution:
    def numPoints(self, darts: List[List[int]], r: int) -> int:
        def distance(x1, y1, x2, y2):
            return sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)

        n = len(darts)
        max_darts = 1

        for i in range(n):
            for j in range(i + 1, n):
                d = distance(darts[i][0], darts[i][1], darts[j][0], darts[j][1])
                if d > 2 * r:
                    continue
                h = sqrt(r ** 2 - (d / 2) ** 2)
                x_mid = (darts[i][0] + darts[j][0]) / 2
                y_mid = (darts[i][1] + darts[j][1]) / 2
                x_diff = (darts[j][1] - darts[i][1]) * h / d
                y_diff = (darts[j][0] - darts[i][0]) * h / d
                center1 = [x_mid + x_diff, y_mid - y_diff]
                center2 = [x_mid - x_diff, y_mid + y_diff]

                for center in [center1, center2]:
                    count = 0
                    for k in range(n):
                        if distance(center[0], center[1], darts[k][0], darts[k][1]) <= r + 1e-6:
                            count += 1
                    max_darts = max(max_darts, count)

        return max_darts

This code will return the maximum number of darts that can lie on the dartboard for the given darts and radius. It iterates through all pairs of darts to find the possible centers of the dartboard and then checks the number of darts that can lie on the dartboard for each center. It returns the maximum number of darts that can lie on the dartboard.

Identifying Problem Isomorphism

“Maximum Number of Darts Inside of a Circular Dartboard” can be isomorphically mapped to “Boats to Save People”.

Reasoning:

In both problems, we are working with a set of points (darts or people) that need to be covered or grouped using some constraints. In the “Maximum Number of Darts Inside of a Circular Dartboard” problem, we are trying to place a dartboard in such a way that the maximum number of darts are within the dartboard. The dartboard can be seen as a group or a container.

In “Boats to Save People”, the boats are the groups or containers, and we want to fit as many people as possible into each boat while not exceeding the boat’s weight limit.

The mapping is not exact as the nature of the two problems are different: one is a geometrical problem and the other is about grouping weights. However, the underlying strategy of grouping based on a certain limit (radius for the dartboard and weight for the boats) makes these two problems conceptually similar.

Both require clever strategies and careful consideration of constraints to solve effectively. However, “Boats to Save People” is simpler because it’s less abstract - you’re working with integer weights and capacities, rather than coordinates on a plane. It also requires less knowledge of geometry and more straightforward array manipulation techniques. If you understand how to solve “Boats to Save People”, you have a good basis for tackling “Maximum Number of Darts Inside of a Circular Dartboard”, though you’ll need to apply some additional geometrical concepts.

10 Prerequisite LeetCode Problems

“1453. Maximum Number of Darts Inside of a Circular Dartboard” involves concepts of geometry, such as the calculation of distances, and the finding of circles covering maximum points. Below are some simpler problems to prepare:

  1. LeetCode 149. Max Points on a Line

    • This problem helps with understanding how to handle points and count maximum points for a condition.

  2. LeetCode 447. Number of Boomerangs

    • This problem deals with the calculation of distances in a 2D plane, similar to the “Maximum Number of Darts Inside of a Circular Dartboard” problem.

  3. LeetCode 611. Valid Triangle Number

    • This problem can help you understand how to deal with geometric shapes in a mathematical way.

  4. LeetCode 792. Number of Matching Subsequences

    • This problem helps with understanding how to count maximum occurrences for a condition.

  5. LeetCode 963. Minimum Area Rectangle II

    • This problem involves the calculation of areas in a 2D plane, which can help to improve understanding of geometry-related problems.

  6. LeetCode 1037. Valid Boomerang

    • This problem involves a simple geometric concept that can help improve spatial reasoning.

  7. LeetCode 1200. Minimum Absolute Difference

    • This problem involves finding a minimum difference, which is a common task in optimization problems like the “Maximum Number of Darts Inside of a Circular Dartboard”.

  8. LeetCode 1266. Minimum Time Visiting All Points

    • This problem also involves calculation of distances in a 2D plane.

  9. LeetCode 1380. Lucky Numbers in a Matrix

    • This problem involves dealing with a 2D structure and finding a maximum condition.

  10. LeetCode 1385. Find the Distance Value Between Two Arrays

    • This problem can help you understand how to handle and calculate distances, which is a key aspect of the “Maximum Number of Darts Inside of a Circular Dartboard” problem.

These cover distances and maximum calculations in a 2D plane, which are key for solving the “1453. Maximum Number of Darts Inside of a Circular Dartboard” problem.