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Approximate Pi from Random Points | Toma Coding Question

Q: Approximate Pi from Random Points

This question evaluates a candidate's understanding of Monte Carlo estimation, basic Euclidean geometry, probability, and numerical reasoning when using random sampling to approximate mathematical constants.

Approximate Pi from Random Points

Company: Toma

Role: Frontend Engineer

Category: Coding & Algorithms

Difficulty: easy

Interview Round: Technical Screen

You are given an array `pts` of points in the unit square. Each point is represented as an array of two numbers `[x, y]`, where `0 <= x <= 1` and `0 <= y <= 1`. A point lies inside the quarter circle of radius `1` if `x*x + y*y <= 1`. Write a function `approx(pts)` that returns an approximation of `pi` using the fraction of points that fall inside the quarter circle. Assume: - `pts` is never `null` - `pts` contains at least one point - every point has exactly two numeric coordinates Example function signature: ```javascript function approx(pts) { return 0; } ```

Quick Answer: This question evaluates a candidate's understanding of Monte Carlo estimation, basic Euclidean geometry, probability, and numerical reasoning when using random sampling to approximate mathematical constants.

You are given an array `pts` of points in the unit square `[0, 1] x [0, 1]`. Each point is represented as `[x, y]`. A point lies inside the quarter circle of radius `1` centered at the origin if `x*x + y*y <= 1`. Use the fraction of points that fall inside the quarter circle to approximate `pi`. Since the area of the quarter circle is `pi / 4` and the area of the unit square is `1`, the approximation is: `pi ≈ 4 * (number of points inside the quarter circle) / (total number of points)` Write a function that returns this approximation as a floating-point number.

Constraints

`1 <= len(pts) <= 100000`
Each point has exactly two numeric coordinates
`0 <= x <= 1` and `0 <= y <= 1` for every point

Examples

Input: [[1, 0]]

Expected Output: 4.0

Explanation: The single point lies exactly on the circle boundary, so it counts as inside. Approximation = 4 * 1 / 1 = 4.0.

Input: [[1, 0], [0, 1], [1, 1], [0.5, 0.5]]

Expected Output: 3.0

Explanation: Three points are inside or on the boundary: [1,0], [0,1], and [0.5,0.5]. One point, [1,1], is outside. Approximation = 4 * 3 / 4 = 3.0.

Input: [[1, 1], [0.9, 0.9]]

Expected Output: 0.0

Explanation: Both points are outside the quarter circle, so the approximation is 4 * 0 / 2 = 0.0.

Input: [[0, 0], [0.6, 0.8], [0.8, 0.6], [0.9, 0.9], [1, 0]]

Expected Output: 3.2

Explanation: Four of the five points are inside or on the boundary: [0,0], [0.6,0.8], [0.8,0.6], and [1,0]. Approximation = 4 * 4 / 5 = 3.2.

Input: [[0.5, 0.5], [0.5, 0.5], [1, 1], [1, 1]]

Expected Output: 2.0

Explanation: The two duplicate [0.5,0.5] points are inside, and the two duplicate [1,1] points are outside. Approximation = 4 * 2 / 4 = 2.0.

Hints

First count how many points satisfy `x*x + y*y <= 1`.
The ratio `inside / total` estimates the area of the quarter circle, which is `pi / 4`.

Quick Overview

This question evaluates a candidate's understanding of Monte Carlo estimation, basic Euclidean geometry, probability, and numerical reasoning when using random sampling to approximate mathematical constants.