1453. Maximum Number of Darts Inside of a Circular Dartboard
Problem Description
Alice has thrown n
darts at positions on a large wall. Each dart's position is given as coordinates [x_i, y_i]
in the array darts
.
Bob wants to place a circular dartboard with radius r
somewhere on the wall to maximize the number of Alice's darts that fall within or on the boundary of the dartboard.
The task is to find the maximum number of darts that can be covered by optimally placing a dartboard of radius r
.
Key Points:
- You have
n
dart positions as 2D coordinates - You need to place a circle of radius
r
to cover as many darts as possible - A dart is covered if it lies inside or exactly on the boundary of the circle
- Return the maximum number of darts that can be covered
Example:
If you have darts at positions [[1,2], [3,4], [5,6]]
and radius r = 2
, you need to find where to place a circle of radius 2 to cover the maximum number of these dart positions.
The solution approach involves:
- For each pair of darts, calculate potential circle centers where a circle of radius
r
passes through both darts - For each potential center, count how many total darts fall within radius
r
- Track the maximum count found
The geometric insight is that an optimal circle placement will have at least two darts on its boundary (unless only one dart can be covered). This is why the solution examines all pairs of darts to find candidate circle centers.
Intuition
The key insight is that if we want to maximize the number of darts covered by a circle, the optimal placement will often have the circle's boundary passing through at least one or two darts. Why? Because if a circle covers some darts but none of them touch its boundary, we could slightly shift the circle to potentially include more darts.
Think about it this way: if we have an optimal circle placement, we can keep moving it until at least two darts are on its boundary. This movement won't reduce the number of covered darts (since we're not pushing any darts outside), and it might even allow us to include more.
This observation drastically reduces our search space. Instead of considering infinite possible positions for the circle center, we only need to consider positions where the circle passes through pairs of darts.
For any two darts within distance 2r
of each other (if they're farther apart, no circle of radius r
can contain both), there are exactly two possible positions for a circle center such that both darts lie on the circle's boundary. These two positions are found by:
- Finding the midpoint between the two darts
- Moving perpendicular to the line connecting the darts by a specific distance
The perpendicular distance is calculated using the Pythagorean theorem: if the distance between two darts is d
, and they both lie on a circle of radius r
, then the center is at distance sqrt(r² - (d/2)²)
from the midpoint, perpendicular to the line connecting the darts.
By checking all pairs of darts and their corresponding possible circle centers, we ensure we don't miss the optimal placement. For each candidate center, we count how many darts fall within radius r
and keep track of the maximum count.
The algorithm handles the edge case where only one dart can be covered by initializing the maximum to 1, and the floating-point comparison uses a small epsilon (1e-7
) to handle numerical precision issues.
Learn more about Math patterns.
Solution Approach
The implementation consists of three main components:
1. Finding Possible Circle Centers (possibleCenters
function)
For two darts at positions (x1, y1)
and (x2, y2)
, we calculate where a circle of radius r
can be centered such that both darts lie on its boundary:
- First, calculate the distance
d
between the two darts:d = sqrt((x2-x1)² + (y2-y1)²)
- If
d > 2r
, no circle of radiusr
can contain both darts, return empty list - Find the midpoint:
(mid_x, mid_y) = ((x1+x2)/2, (y1+y2)/2)
- Calculate the perpendicular distance from midpoint to center:
dist_to_center = sqrt(r² - (d/2)²)
- The perpendicular direction is obtained by rotating the vector
(dx, dy)
by 90 degrees:(dy, -dx)
- Normalize and scale this perpendicular vector to get the offset:
offset_x = dist_to_center * dy / d
offset_y = dist_to_center * -dx / d
- Return two possible centers: one on each side of the line connecting the darts
2. Counting Darts Within Circle (countDarts
function)
For a given center (x, y)
, count how many darts fall within or on the circle:
- Iterate through all darts
- Calculate distance from each dart to the center using the distance formula
- If distance ≤
r + 1e-7
, increment the count (the small epsilon handles floating-point precision)
3. Main Algorithm
The main algorithm exhaustively checks all possible optimal placements:
max_darts = 1 # At least one dart can always be covered
for i in range(n):
for j in range(i + 1, n):
# Get possible centers where circle passes through darts i and j
centers = possibleCenters(darts[i], darts[j])
# For each center, count darts and update maximum
for center in centers:
max_darts = max(max_darts, countDarts(center))
Time Complexity: O(n² × n) = O(n³)
where n
is the number of darts
- We check
O(n²)
pairs of darts - For each pair, we generate at most 2 centers
- For each center, we count darts in
O(n)
time
Space Complexity: O(1)
as we only use a constant amount of extra space regardless of input size
The algorithm guarantees finding the optimal solution because it checks all geometrically significant positions where the circle boundary passes through pairs of darts, which includes all potential optimal placements.
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Start EvaluatorExample Walkthrough
Let's walk through a concrete example with darts at positions [[1,2], [3,2], [2,4]]
and radius r = 2
.
Step 1: Initialize
- Start with
max_darts = 1
(we can always cover at least one dart)
Step 2: Check all pairs of darts
Pair 1: Darts [1,2] and [3,2]
- Distance between them:
d = sqrt((3-1)² + (2-2)²) = 2
- Since
d = 2 ≤ 2r = 4
, we can find circle centers - Midpoint:
(2, 2)
- Perpendicular distance:
sqrt(4 - 1) = sqrt(3) ≈ 1.732
- The line between darts is horizontal, so perpendicular is vertical
- Two possible centers:
(2, 2+1.732)
and(2, 2-1.732)
- Center at
(2, 3.732)
: Check distances to all darts- Dart [1,2]:
sqrt((1-2)² + (2-3.732)²) = sqrt(1 + 3) = 2
✓ (on boundary) - Dart [3,2]:
sqrt((3-2)² + (2-3.732)²) = sqrt(1 + 3) = 2
✓ (on boundary) - Dart [2,4]:
sqrt((2-2)² + (4-3.732)²) = 0.268 < 2
✓ (inside) - Count: 3 darts
- Dart [1,2]:
- Center at
(2, 0.268)
: Would only cover the first two darts
- Center at
Pair 2: Darts [1,2] and [2,4]
- Distance:
d = sqrt(1 + 4) = sqrt(5) ≈ 2.236
- Since
d ≤ 4
, find centers - Midpoint:
(1.5, 3)
- Perpendicular distance:
sqrt(4 - 1.25) ≈ 1.658
- Calculate two centers and count covered darts
- Best count from this pair: 2 darts
Pair 3: Darts [3,2] and [2,4]
- Distance:
d = sqrt(1 + 4) = sqrt(5) ≈ 2.236
- Similar process yields at most 2 darts covered
Step 3: Return maximum
- The maximum count found is 3 darts, achieved when the circle is centered at approximately
(2, 3.732)
This example demonstrates how the algorithm systematically checks all pairs of darts to find circle placements where two darts lie on the boundary, then counts all darts within each such circle to find the optimal placement.
Solution Implementation
1class Solution:
2 def numPoints(self, darts: list[list[int]], r: int) -> int:
3 """
4 Find the maximum number of darts that can be enclosed within a dartboard of radius r.
5
6 The algorithm works by finding all possible circle centers that pass through pairs of darts,
7 then counting how many darts fall within each circle.
8
9 Args:
10 darts: List of [x, y] coordinates representing dart positions
11 r: Radius of the dartboard
12
13 Returns:
14 Maximum number of darts that can be enclosed in a circle of radius r
15 """
16 from math import sqrt
17
18 def count_darts_in_circle(center_x: float, center_y: float) -> int:
19 """
20 Count how many darts are within the circle centered at (center_x, center_y).
21
22 Args:
23 center_x: X-coordinate of circle center
24 center_y: Y-coordinate of circle center
25
26 Returns:
27 Number of darts within the circle (including boundary)
28 """
29 count = 0
30 for dart_x, dart_y in darts:
31 # Calculate distance from dart to center
32 distance = sqrt((center_x - dart_x) ** 2 + (center_y - dart_y) ** 2)
33 # Check if dart is within circle (with small epsilon for floating point comparison)
34 if distance <= r + 1e-7:
35 count += 1
36 return count
37
38 def find_circle_centers(x1: float, y1: float, x2: float, y2: float) -> list[tuple[float, float]]:
39 """
40 Find the two possible centers of circles with radius r that pass through
41 points (x1, y1) and (x2, y2).
42
43 Args:
44 x1, y1: Coordinates of first point
45 x2, y2: Coordinates of second point
46
47 Returns:
48 List of up to 2 possible circle centers as (x, y) tuples
49 """
50 # Calculate distance between the two points
51 dx = x2 - x1
52 dy = y2 - y1
53 distance_between_points = sqrt(dx * dx + dy * dy)
54
55 # If points are too far apart, no circle of radius r can pass through both
56 if distance_between_points > 2 * r:
57 return []
58
59 # Find midpoint between the two points
60 mid_x = (x1 + x2) / 2
61 mid_y = (y1 + y2) / 2
62
63 # Calculate distance from midpoint to circle center
64 # Using Pythagorean theorem: r^2 = (d/2)^2 + dist_to_center^2
65 dist_to_center = sqrt(r * r - (distance_between_points / 2) * (distance_between_points / 2))
66
67 # Calculate perpendicular offset from midpoint to circle centers
68 # The perpendicular direction is obtained by rotating the vector (dx, dy) by 90 degrees
69 offset_x = dist_to_center * dy / distance_between_points
70 offset_y = dist_to_center * (-dx) / distance_between_points
71
72 # Return both possible circle centers
73 return [
74 (mid_x + offset_x, mid_y + offset_y),
75 (mid_x - offset_x, mid_y - offset_y),
76 ]
77
78 n = len(darts)
79 max_darts_in_circle = 1 # At minimum, we can always enclose one dart
80
81 # Try all pairs of darts to find circle centers
82 for i in range(n):
83 for j in range(i + 1, n):
84 # Find possible circle centers that pass through darts[i] and darts[j]
85 possible_centers = find_circle_centers(
86 darts[i][0], darts[i][1],
87 darts[j][0], darts[j][1]
88 )
89
90 # For each possible center, count darts and update maximum
91 for center_x, center_y in possible_centers:
92 darts_count = count_darts_in_circle(center_x, center_y)
93 max_darts_in_circle = max(max_darts_in_circle, darts_count)
94
95 return max_darts_in_circle
96
1class Solution {
2 /**
3 * Finds the maximum number of darts that can be enclosed within a circle of radius r.
4 * Uses the property that the optimal circle must pass through at least two dart points.
5 *
6 * @param darts 2D array where each element represents [x, y] coordinates of a dart
7 * @param r radius of the circle
8 * @return maximum number of darts that can be enclosed
9 */
10 public int numPoints(int[][] darts, int r) {
11 int n = darts.length;
12 int maxDarts = 1; // At minimum, we can always enclose one dart
13
14 // Try all pairs of darts as potential points on the circle boundary
15 for (int i = 0; i < n; i++) {
16 for (int j = i + 1; j < n; j++) {
17 // Find possible circle centers that pass through both darts[i] and darts[j]
18 List<double[]> centers = possibleCenters(
19 darts[i][0], darts[i][1],
20 darts[j][0], darts[j][1],
21 r
22 );
23
24 // For each possible center, count how many darts it can enclose
25 for (double[] center : centers) {
26 maxDarts = Math.max(maxDarts, countDarts(center[0], center[1], darts, r));
27 }
28 }
29 }
30
31 return maxDarts;
32 }
33
34 /**
35 * Calculates possible circle centers with radius r that pass through two given points.
36 * There can be 0, 1, or 2 such centers depending on the distance between points.
37 *
38 * @param x1, y1 coordinates of first point
39 * @param x2, y2 coordinates of second point
40 * @param r radius of the circle
41 * @return list of possible center coordinates [x, y]
42 */
43 private List<double[]> possibleCenters(int x1, int y1, int x2, int y2, int r) {
44 List<double[]> centers = new ArrayList<>();
45
46 // Calculate distance between the two points
47 double dx = x2 - x1;
48 double dy = y2 - y1;
49 double distance = Math.sqrt(dx * dx + dy * dy);
50
51 // If points are too far apart, no circle of radius r can pass through both
52 if (distance > 2 * r) {
53 return centers;
54 }
55
56 // Find midpoint between the two points
57 double midX = (x1 + x2) / 2.0;
58 double midY = (y1 + y2) / 2.0;
59
60 // Calculate perpendicular distance from midpoint to circle center
61 // Using Pythagorean theorem: r² = (d/2)² + distToCenter²
62 double distToCenter = Math.sqrt(r * r - (distance / 2.0) * (distance / 2.0));
63
64 // Calculate perpendicular offset direction (rotated 90 degrees)
65 // Perpendicular vector to (dx, dy) is (-dy, dx) or (dy, -dx)
66 double offsetX = distToCenter * dy / distance;
67 double offsetY = distToCenter * (-dx) / distance;
68
69 // Two possible centers: one on each side of the line connecting the points
70 centers.add(new double[] {midX + offsetX, midY + offsetY});
71 centers.add(new double[] {midX - offsetX, midY - offsetY});
72
73 return centers;
74 }
75
76 /**
77 * Counts how many darts are within or on the boundary of a circle.
78 *
79 * @param x, y coordinates of the circle center
80 * @param darts array of dart coordinates
81 * @param r radius of the circle
82 * @return number of darts enclosed by the circle
83 */
84 private int countDarts(double x, double y, int[][] darts, int r) {
85 int count = 0;
86 double epsilon = 1e-7; // Small tolerance for floating-point comparison
87
88 for (int[] dart : darts) {
89 // Calculate distance from dart to circle center
90 double distance = Math.sqrt(
91 Math.pow(dart[0] - x, 2) + Math.pow(dart[1] - y, 2)
92 );
93
94 // Check if dart is within or on the circle boundary (with tolerance)
95 if (distance <= r + epsilon) {
96 count++;
97 }
98 }
99
100 return count;
101 }
102}
103
1class Solution {
2public:
3 /**
4 * Finds the maximum number of darts that can be enclosed within a circle of radius r.
5 * Uses the property that the optimal circle must pass through at least two dart points.
6 *
7 * @param darts 2D vector where each element represents [x, y] coordinates of a dart
8 * @param r radius of the circle
9 * @return maximum number of darts that can be enclosed
10 */
11 int numPoints(vector<vector<int>>& darts, int r) {
12 int n = darts.size();
13 int maxDarts = 1; // At minimum, we can always enclose one dart
14
15 // Try all pairs of darts as potential points on the circle boundary
16 for (int i = 0; i < n; i++) {
17 for (int j = i + 1; j < n; j++) {
18 // Find possible circle centers that pass through both darts[i] and darts[j]
19 vector<pair<double, double>> centers = possibleCenters(
20 darts[i][0], darts[i][1],
21 darts[j][0], darts[j][1],
22 r
23 );
24
25 // For each possible center, count how many darts it can enclose
26 for (const auto& center : centers) {
27 maxDarts = max(maxDarts, countDarts(center.first, center.second, darts, r));
28 }
29 }
30 }
31
32 return maxDarts;
33 }
34
35private:
36 /**
37 * Calculates possible circle centers with radius r that pass through two given points.
38 * There can be 0, 1, or 2 such centers depending on the distance between points.
39 *
40 * @param x1, y1 coordinates of first point
41 * @param x2, y2 coordinates of second point
42 * @param r radius of the circle
43 * @return vector of possible center coordinates as pairs (x, y)
44 */
45 vector<pair<double, double>> possibleCenters(int x1, int y1, int x2, int y2, int r) {
46 vector<pair<double, double>> centers;
47
48 // Calculate distance between the two points
49 double dx = x2 - x1;
50 double dy = y2 - y1;
51 double distance = sqrt(dx * dx + dy * dy);
52
53 // If points are too far apart, no circle of radius r can pass through both
54 if (distance > 2 * r) {
55 return centers;
56 }
57
58 // Find midpoint between the two points
59 double midX = (x1 + x2) / 2.0;
60 double midY = (y1 + y2) / 2.0;
61
62 // Special case: if points are the same, return the point itself as center
63 if (distance < 1e-7) {
64 centers.push_back({static_cast<double>(x1), static_cast<double>(y1)});
65 return centers;
66 }
67
68 // Calculate perpendicular distance from midpoint to circle center
69 // Using Pythagorean theorem: r² = (d/2)² + distToCenter²
70 double distToCenter = sqrt(r * r - (distance / 2.0) * (distance / 2.0));
71
72 // Calculate perpendicular offset direction (rotated 90 degrees)
73 // Perpendicular vector to (dx, dy) is (-dy, dx) or (dy, -dx)
74 double offsetX = distToCenter * dy / distance;
75 double offsetY = distToCenter * (-dx) / distance;
76
77 // Two possible centers: one on each side of the line connecting the points
78 centers.push_back({midX + offsetX, midY + offsetY});
79 centers.push_back({midX - offsetX, midY - offsetY});
80
81 return centers;
82 }
83
84 /**
85 * Counts how many darts are within or on the boundary of a circle.
86 *
87 * @param x, y coordinates of the circle center
88 * @param darts vector of dart coordinates
89 * @param r radius of the circle
90 * @return number of darts enclosed by the circle
91 */
92 int countDarts(double x, double y, const vector<vector<int>>& darts, int r) {
93 int count = 0;
94 double epsilon = 1e-7; // Small tolerance for floating-point comparison
95
96 for (const auto& dart : darts) {
97 // Calculate distance from dart to circle center
98 double distance = sqrt(
99 pow(dart[0] - x, 2) + pow(dart[1] - y, 2)
100 );
101
102 // Check if dart is within or on the circle boundary (with tolerance)
103 if (distance <= r + epsilon) {
104 count++;
105 }
106 }
107
108 return count;
109 }
110};
111
1/**
2 * Finds the maximum number of darts that can be enclosed within a circle of radius r.
3 * Uses the property that the optimal circle must pass through at least two dart points.
4 *
5 * @param darts 2D array where each element represents [x, y] coordinates of a dart
6 * @param r radius of the circle
7 * @return maximum number of darts that can be enclosed
8 */
9function numPoints(darts: number[][], r: number): number {
10 const n = darts.length;
11 let maxDarts = 1; // At minimum, we can always enclose one dart
12
13 // Try all pairs of darts as potential points on the circle boundary
14 for (let i = 0; i < n; i++) {
15 for (let j = i + 1; j < n; j++) {
16 // Find possible circle centers that pass through both darts[i] and darts[j]
17 const centers = possibleCenters(
18 darts[i][0], darts[i][1],
19 darts[j][0], darts[j][1],
20 r
21 );
22
23 // For each possible center, count how many darts it can enclose
24 for (const center of centers) {
25 maxDarts = Math.max(maxDarts, countDarts(center[0], center[1], darts, r));
26 }
27 }
28 }
29
30 return maxDarts;
31}
32
33/**
34 * Calculates possible circle centers with radius r that pass through two given points.
35 * There can be 0, 1, or 2 such centers depending on the distance between points.
36 *
37 * @param x1 x-coordinate of first point
38 * @param y1 y-coordinate of first point
39 * @param x2 x-coordinate of second point
40 * @param y2 y-coordinate of second point
41 * @param r radius of the circle
42 * @return array of possible center coordinates [x, y]
43 */
44function possibleCenters(x1: number, y1: number, x2: number, y2: number, r: number): number[][] {
45 const centers: number[][] = [];
46
47 // Calculate distance between the two points
48 const dx = x2 - x1;
49 const dy = y2 - y1;
50 const distance = Math.sqrt(dx * dx + dy * dy);
51
52 // If points are too far apart, no circle of radius r can pass through both
53 if (distance > 2 * r) {
54 return centers;
55 }
56
57 // Find midpoint between the two points
58 const midX = (x1 + x2) / 2.0;
59 const midY = (y1 + y2) / 2.0;
60
61 // Special case: if two points are the same, return the point as center
62 if (distance === 0) {
63 centers.push([midX, midY]);
64 return centers;
65 }
66
67 // Calculate perpendicular distance from midpoint to circle center
68 // Using Pythagorean theorem: r² = (d/2)² + distToCenter²
69 const distToCenter = Math.sqrt(r * r - (distance / 2.0) * (distance / 2.0));
70
71 // Calculate perpendicular offset direction (rotated 90 degrees)
72 // Perpendicular vector to (dx, dy) is (-dy, dx) or (dy, -dx)
73 const offsetX = distToCenter * dy / distance;
74 const offsetY = distToCenter * (-dx) / distance;
75
76 // Two possible centers: one on each side of the line connecting the points
77 centers.push([midX + offsetX, midY + offsetY]);
78 centers.push([midX - offsetX, midY - offsetY]);
79
80 return centers;
81}
82
83/**
84 * Counts how many darts are within or on the boundary of a circle.
85 *
86 * @param x x-coordinate of the circle center
87 * @param y y-coordinate of the circle center
88 * @param darts array of dart coordinates
89 * @param r radius of the circle
90 * @return number of darts enclosed by the circle
91 */
92function countDarts(x: number, y: number, darts: number[][], r: number): number {
93 let count = 0;
94 const epsilon = 1e-7; // Small tolerance for floating-point comparison
95
96 for (const dart of darts) {
97 // Calculate distance from dart to circle center
98 const distance = Math.sqrt(
99 Math.pow(dart[0] - x, 2) + Math.pow(dart[1] - y, 2)
100 );
101
102 // Check if dart is within or on the circle boundary (with tolerance)
103 if (distance <= r + epsilon) {
104 count++;
105 }
106 }
107
108 return count;
109}
110
Time and Space Complexity
Time Complexity: O(n³)
where n
is the number of darts.
The analysis breaks down as follows:
- The outer two nested loops iterate through all pairs of darts:
O(n²)
pairs in total - For each pair of darts,
possibleCenters
computes at most 2 circle centers inO(1)
time - For each center (at most 2 per pair),
countDarts
is called, which iterates through alln
darts to count how many fall within the radius:O(n)
time - Overall:
O(n²) × O(1) × 2 × O(n) = O(n³)
Space Complexity: O(1)
The space analysis:
- The
possibleCenters
function creates a list with at most 2 coordinate pairs:O(1)
- The
countDarts
function uses only a counter variable:O(1)
- No recursive calls or data structures that scale with input size
- Only a constant amount of variables (
max_darts
,centers
, loop indices, temporary calculations) are used - Overall auxiliary space:
O(1)
Note: The input array darts
is not counted in space complexity as it's part of the input.
Learn more about how to find time and space complexity quickly.
Common Pitfalls
1. Floating-Point Precision Errors
The Problem: When checking if a dart is within the circle, direct floating-point comparison can fail due to precision errors. A dart that should be exactly on the boundary might be calculated as slightly outside due to accumulated floating-point arithmetic errors.
# Problematic code: if distance <= r: # May incorrectly exclude boundary points count += 1
The Solution: Add a small epsilon value to account for floating-point imprecision:
# Correct approach: if distance <= r + 1e-7: # Small epsilon handles precision issues count += 1
2. Division by Zero When Points Coincide
The Problem: When two darts are at the exact same position (distance = 0), the normalization step divides by zero:
# This causes division by zero when distance_between_points = 0: offset_x = dist_to_center * dy / distance_between_points offset_y = dist_to_center * (-dx) / distance_between_points
The Solution: Check for coincident points before calculating centers:
def find_circle_centers(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
distance_between_points = sqrt(dx * dx + dy * dy)
# Handle coincident points
if distance_between_points < 1e-9:
# Return the point itself as the only center
return [(x1, y1)]
# Continue with normal calculation...
if distance_between_points > 2 * r:
return []
3. Missing Edge Case: Single Dart Coverage
The Problem: The algorithm only checks circles passing through pairs of darts, but might miss the optimal placement when the best solution is a circle centered directly on a single dart (with no other darts on the boundary).
The Solution: Additionally check circles centered at each dart position:
# Add this to the main algorithm:
for i in range(n):
# Check circle centered at dart i
count = count_darts_in_circle(darts[i][0], darts[i][1])
max_darts_in_circle = max(max_darts_in_circle, count)
4. Numerical Instability in Square Root Calculation
The Problem:
When two points are very close to being exactly 2r apart, the expression r * r - (distance_between_points / 2) * (distance_between_points / 2)
can become slightly negative due to floating-point errors, causing sqrt()
to fail.
The Solution: Clamp the value to ensure it's non-negative:
# Instead of:
dist_to_center = sqrt(r * r - (distance_between_points / 2) * (distance_between_points / 2))
# Use:
squared_dist = r * r - (distance_between_points / 2) * (distance_between_points / 2)
dist_to_center = sqrt(max(0, squared_dist)) # Clamp to non-negative
5. Not Handling the Case Where All Darts Are Far Apart
The Problem: When all darts are more than 2r apart from each other, no pair will generate valid circle centers, but the algorithm should still return 1 (covering at least one dart).
The Solution:
Initialize max_darts_in_circle = 1
at the start, which handles this case automatically, or explicitly check circles centered at each individual dart as mentioned in pitfall #3.
How does quick sort divide the problem into subproblems?
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