Find the Child Who Has the Ball After K Seconds - Problem

Math Simulation Easy

You are given two positive integers n and k. There are n children numbered from 0 to n - 1 standing in a queue in order from left to right.

Initially, child 0 holds a ball and the direction of passing the ball is towards the right direction. After each second, the child holding the ball passes it to the child next to them. Once the ball reaches either end of the line, i.e. child 0 or child n - 1, the direction of passing is reversed.

Return the number of the child who receives the ball after k seconds.

Input & Output

Example 1 — Basic Case

$ Input: n = 3, k = 5

› Output: 1

💡 Note: Ball moves: 0→1→2→1→0→1. After 5 seconds, ball is at position 1.

Example 2 — Reaches End

$ Input: n = 5, k = 6

› Output: 2

💡 Note: Ball moves: 0→1→2→3→4→3→2. After 6 seconds, ball is at position 2.

Example 3 — Single Child

$ Input: n = 1, k = 10

› Output: 0

💡 Note: Only one child, so ball always stays at position 0.

Constraints

1 ≤ n ≤ 50
1 ≤ k ≤ 50

Visualization

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Asked in

G Google 15 a Amazon 12 M Microsoft 8

The key insight is recognizing the zigzag pattern that repeats every 2(n-1) seconds. Best approach uses mathematical formula for O(1) solution. Time: O(1), Space: O(1)

Common Approaches

✓ Mathematical Pattern

⏱️ Time: O(1) Space: O(1)

The ball follows a zigzag pattern that repeats every 2(n-1) seconds. We can use this cycle to calculate the final position without simulation.

Simulation

⏱️ Time: O(k) Space: O(1)

Track the current position and direction, then simulate each second by moving the ball to the next child and reversing direction when hitting the ends.

Mathematical Pattern — Algorithm Steps

Step 1: Calculate cycle length as 2(n-1)
Step 2: Find position within cycle using k % cycle
Step 3: Map cycle position to actual child position

Visualization

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Step-by-Step Walkthrough

Identify Pattern

Ball moves 0→1→2→1→0→1→2... in cycles

Calculate Cycle

Cycle length = 2(n-1) = 4 for n=3

Direct Lookup

Use k % cycle_length to find position

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int solution(int n, int k) {
    if (n == 1) return 0;
    
    // The ball follows a cycle of length 2(n-1)
    int cycle_length = 2 * (n - 1);
    int k_mod = k % cycle_length;
    
    if (k_mod <= n - 1) {
        return k_mod;
    } else {
        return 2 * (n - 1) - k_mod;
    }
}

int main() {
    int n, k;
    scanf("%d", &n);
    scanf("%d", &k);
    printf("%d\n", solution(n, k));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Direct calculation using modular arithmetic

✓ Linear Growth

Space Complexity

O(1)

Only need a few variables for calculation

✓ Linear Space

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Find the Child Who Has the Ball After K Seconds - Problem

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Mathematical Pattern — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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