#3178

Easy

LeetCode Problem Workspace

Find the Child Who Has the Ball After K Seconds

Find the child who holds the ball after k seconds of passing in a queue, considering reversals at both ends.

Problem Statement

You are given two integers, n and k. n represents the number of children standing in a queue, numbered from 0 to n-1, and initially, child 0 holds the ball. The ball moves to the right, passing to the next child every second. If the ball reaches the end of the queue, the direction reverses, passing it back towards the starting point. The process repeats every time the ball reaches one end of the queue.

The task is to determine which child will have the ball after exactly k seconds, taking into account the back-and-forth motion of the ball. If k is large enough, the ball may pass through multiple cycles before settling on the final child.

Examples

Example 1

Input: n = 3, k = 5

Output: 1

Example 2

Input: n = 5, k = 6

Output: 2

Example 3

Input: n = 4, k = 2

Output: 2

Constraints

2 <= n <= 50
1 <= k <= 50

Solution Approach

Simulation of Ball Passing

Simulate each second of the process by keeping track of the current position of the ball and whether it is moving forward or backward. At each step, adjust the position accordingly and reverse the direction if the ball reaches the ends of the queue.

Modulo Arithmetic Optimization

Instead of simulating each second for large values of k, use modulo arithmetic to reduce the problem to a smaller equivalent. The ball will return to the starting point after 2 * (n-1) seconds. Thus, you can calculate the effective time within a cycle of this length.

Pattern Identification

After observing the ball’s movement, recognize that after 2 * (n-1) seconds, the sequence starts repeating. This pattern can be used to find the child holding the ball without explicitly simulating every second.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach chosen. A direct simulation approach runs in O(k), while a more optimized approach using modulo arithmetic or pattern recognition reduces the complexity to O(1). The space complexity remains O(1) for both approaches.

What Interviewers Usually Probe

Can the candidate identify the repeating pattern in the ball's movement?
Does the candidate choose an optimized solution over brute force simulation?
Can the candidate handle the problem efficiently with larger k values?

Common Pitfalls or Variants

Common pitfalls

Not recognizing the pattern in the ball's movement and resorting to unnecessary simulations.
Failing to account for the direction reversal at the ends of the queue.
Incorrectly handling small values of k, leading to off-by-one errors.

Follow-up variants

Consider larger values of n or k for performance testing.
Adjust the problem by having the ball start at a different position, or have different patterns of movement.
Implement the simulation with different boundary conditions, such as multiple reversals.

FAQ

What is the main challenge in solving the 'Find the Child Who Has the Ball After K Seconds' problem?

The challenge is recognizing the pattern in the ball's movement and optimizing the solution using modulo arithmetic to avoid simulating every second.

How can I optimize my solution for large values of k?

You can optimize the solution by using modulo arithmetic to map the problem into a smaller cycle, reducing unnecessary simulations.

What are the key patterns to notice in this problem?

Key patterns include the ball reversing direction at the ends of the queue and the fact that the ball returns to the starting point after 2 * (n - 1) seconds.

How does GhostInterview help with this problem?

GhostInterview provides detailed solutions and optimizations to efficiently solve this problem, including identifying patterns and reducing complexity.

Can the simulation approach lead to inefficiencies?

Yes, a direct simulation can be inefficient for large k values. Using modulo arithmetic or pattern identification can optimize the solution significantly.

terminal

Solution

Solution 1: Mathematics

We notice that there are $n - 1$ passes in each round. Therefore, we can take $k$ modulo $n - 1$ to get the number of passes $mod$ in the current round. Then we divide $k$ by $n - 1$ to get the current round number $k$.

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class Solution:
    def numberOfChild(self, n: int, k: int) -> int:
        k, mod = divmod(k, n - 1)
        return n - mod - 1 if k & 1 else mod

Continue Practicing

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