Mirror Reflection

Mirror Reflection - Problem

There is a special square room with mirrors on each of the four walls. Except for the southwest corner, there are receptors on each of the remaining corners, numbered 0, 1, and 2.

The square room has walls of length p and a laser ray from the southwest corner first meets the east wall at a distance q from the 0th receptor.

Given the two integers p and q, return the number of the receptor that the ray meets first.

The test cases are guaranteed so that the ray will meet a receptor eventually.

Input & Output

Example 1 — Basic Case

$ Input: p = 2, q = 1

› Output: 2

💡 Note: The ray starts at (0,0) and first hits the east wall at (2,1). After bouncing, it eventually reaches receptor 2 at the northwest corner (0,2).

Example 2 — Different Path

$ Input: p = 3, q = 1

› Output: 1

💡 Note: The ray bounces multiple times but eventually hits receptor 1 at the northeast corner (3,3).

Example 3 — Direct Hit

$ Input: p = 1, q = 1

› Output: 1

💡 Note: The ray travels diagonally and directly hits receptor 1 at (1,1) without any bounces.

Constraints

1 ≤ p ≤ 1000
0 < q < p

Visualization

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Asked in

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The key insight is that instead of simulating the laser bounces, we can use mathematical properties. After reducing p and q by their GCD, the parity (even/odd) of the reduced values determines which receptor is hit. Best approach uses GCD calculation. Time: O(log(min(p,q))), Space: O(1)

Common Approaches

✓ Mathematical GCD Approach

⏱️ Time: O(log(min(p,q))) Space: O(1)

Instead of simulation, use the fact that the ray hits a receptor when we reach LCM(p,q). Use GCD properties to determine the final position without simulation.

Simulation Approach

⏱️ Time: O(p+q) Space: O(1)

Track the laser's position and direction, bouncing off walls until reaching a corner with a receptor. Calculate new position after each wall collision.

Mathematical GCD Approach — Algorithm Steps

Calculate LCM(p,q) = p*q/GCD(p,q)
Determine final coordinates using LCM
Map coordinates to receptor number based on position

Visualization

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Step-by-Step Walkthrough

Reduce Fractions

Divide p,q by GCD to get simplest form

Check Parity

Use even/odd properties to determine receptor

Direct Answer

Map parity to receptor number

Code -

solution.c — C

#include <stdio.h>

int gcd(int a, int b) {
    while (b != 0) {
        int temp = b;
        b = a % b;
        a = temp;
    }
    return a;
}

int solution(int p, int q) {
    int g = gcd(p, q);
    p /= g;
    q /= g;
    
    if (p % 2 == 0) {
        return q % 2 == 1 ? 2 : 0;
    } else {
        return 1;
    }
}

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

Time & Space Complexity

Time Complexity

⏱️

O(log(min(p,q)))

GCD calculation using Euclidean algorithm

✓ Linear Growth

Space Complexity

O(1)

Only storing a few integer variables

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Mathematical GCD Approach — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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