LeetCode #447 — MEDIUM

Number of Boomerangs

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

You are given n points in the plane that are all distinct, where points[i] = [x_i, y_i]. A boomerang is a tuple of points (i, j, k) such that the distance between i and j equals the distance between i and k (the order of the tuple matters).

Return the number of boomerangs.

Example 1:

Input: points = [[0,0],[1,0],[2,0]]
Output: 2
Explanation: The two boomerangs are [[1,0],[0,0],[2,0]] and [[1,0],[2,0],[0,0]].

Example 2:

Input: points = [[1,1],[2,2],[3,3]]
Output: 2

Example 3:

Input: points = [[1,1]]
Output: 0

Constraints:

n == points.length
1 <= n <= 500
points[i].length == 2
-10⁴ <= x_i, y_i <= 10⁴
All the points are unique.

Step 01

Brute Force Baseline

Problem summary: You are given n points in the plane that are all distinct, where points[i] = [xi, yi]. A boomerang is a tuple of points (i, j, k) such that the distance between i and j equals the distance between i and k (the order of the tuple matters). Return the number of boomerangs.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Hash Map · Math

Example 1

[[0,0],[1,0],[2,0]]

Example 2

[[1,1],[2,2],[3,3]]

Example 3

[[1,1]]

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #447: Number of Boomerangs
class Solution {
    public int numberOfBoomerangs(int[][] points) {
        int ans = 0;
        for (int[] p1 : points) {
            Map<Integer, Integer> cnt = new HashMap<>();
            for (int[] p2 : points) {
                int d = (p1[0] - p2[0]) * (p1[0] - p2[0]) + (p1[1] - p2[1]) * (p1[1] - p2[1]);
                ans += cnt.getOrDefault(d, 0);
                cnt.merge(d, 1, Integer::sum);
            }
        }
        return ans << 1;
    }
}

// Accepted solution for LeetCode #447: Number of Boomerangs
func numberOfBoomerangs(points [][]int) (ans int) {
	for _, p1 := range points {
		cnt := map[int]int{}
		for _, p2 := range points {
			d := (p1[0]-p2[0])*(p1[0]-p2[0]) + (p1[1]-p2[1])*(p1[1]-p2[1])
			ans += cnt[d]
			cnt[d]++
		}
	}
	ans <<= 1
	return
}

# Accepted solution for LeetCode #447: Number of Boomerangs
class Solution:
    def numberOfBoomerangs(self, points: List[List[int]]) -> int:
        ans = 0
        for p1 in points:
            cnt = Counter()
            for p2 in points:
                d = dist(p1, p2)
                ans += cnt[d]
                cnt[d] += 1
        return ans << 1

// Accepted solution for LeetCode #447: Number of Boomerangs
struct Solution;

use std::collections::HashMap;

impl Solution {
    fn distance_square(a: &[i32], b: &[i32]) -> i32 {
        (a[0] - b[0]) * (a[0] - b[0]) + (a[1] - b[1]) * (a[1] - b[1])
    }
    fn number_of_boomerangs(points: Vec<Vec<i32>>) -> i32 {
        let n = points.len();
        let mut hm: HashMap<i32, i32> = HashMap::new();
        let mut sum = 0;
        for i in 0..n {
            for j in 0..n {
                if i == j {
                    continue;
                }
                let a = &points[i];
                let b = &points[j];
                let distance_square = Self::distance_square(a, b);
                let ea = hm.entry(distance_square).or_default();
                *ea += 1;
            }
            for &value in hm.values() {
                if value > 1 {
                    sum += value * (value - 1);
                }
            }
            hm.clear();
        }
        sum
    }
}

#[test]
fn test() {
    let points: Vec<Vec<i32>> = vec_vec_i32![[0, 0], [1, 0], [2, 0]];
    assert_eq!(Solution::number_of_boomerangs(points), 2);
}

// Accepted solution for LeetCode #447: Number of Boomerangs
function numberOfBoomerangs(points: number[][]): number {
    let ans = 0;
    for (const [x1, y1] of points) {
        const cnt: Map<number, number> = new Map();
        for (const [x2, y2] of points) {
            const d = (x1 - x2) ** 2 + (y1 - y2) ** 2;
            ans += cnt.get(d) || 0;
            cnt.set(d, (cnt.get(d) || 0) + 1);
        }
    }
    return ans << 1;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n^2)

Space

O(n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.