Two Balls Reachability Game

Given A numbers of white and B numbers of black balls. You need to have X number of white and Y number of black balls (A <= X, B <= Y) to win the game by doing some(zero or more) operations.
In one operation: At any moment if you have p white and q black balls then at that moment you can buy q white or p black balls.
Find if it’s possible to have X white and Y black balls at the end or not. 
Examples:

Input:  A = 1, B = 1, X = 3, Y = 8
Output: POSSIBLE

Explanation:
The steps are, (1, 1)->(1, 2)->(3, 2)->(3, 5)->(3, 8)

Input: A = 3, Y = 2, X = 4, Y = 6
Output: NOT POSSIBLE

Approach:
We have to solve this problem using the property of gcd. Let’s see how. 

1. Initially, we have A white and B black ball. We have to get rest of the X-A Red balls and Y-B Black balls. 
   
2. Below is the property of gcd of two numbers that we will use, 
   

gcd(x, y) = gcd(x + y, y)
gcd(x, y) = gcd(x, y + x)

2. This property is the same as the operation which is mentioned in the question. So from here, we get that if the gcd of the final state is the same as gcd of initial state then it is always possible to reach the goal, otherwise not. 
   

Below is the implementation of the above approach: 

C++

// C++ program to Find is it possible
// to have X white and Y black
// balls at the end.
#include <bits/stdc++.h>
using namespace std;

// Recursive function to return
// gcd of a and b
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}

// Function returns if it's
// possible to have X white
// and Y black balls or not.
void IsPossible(int a, int b,
                int x, int y)
{

    // Finding gcd of (x, y)
    // and (a, b)
    int final = gcd(x, y);
    int initial = gcd(a, b);

     // If gcd is same, it's always
    // possible to reach (x, y)
    if (initial == final)
    {

        cout << "POSSIBLE\n";
    }
    else
    {
        // Here it's never possible
        // if gcd is not same
        cout << "NOT POSSIBLE\n";
    }
}

// Driver Code
int main()
{

    int A = 1, B = 2, X = 4, Y = 11;
    IsPossible(A, B, X, Y);

    A = 2, B = 2, X = 3, Y = 6;
    IsPossible(A, B, X, Y);

    return 0;
}

Java

// Java program to Find is it possible
// to have X white and Y black
// balls at the end.
import java.io.*;

class GFG{

// Recursive function to return
// gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}

// Function returns if it's
// possible to have X white
// and Y black balls or not.
static void IsPossible(int a, int b,
                       int x, int y)
{

    // Finding gcd of (x, y)
    // and (a, b)
    int g = gcd(x, y);
    int initial = gcd(a, b);

    // If gcd is same, it's always
    // possible to reach (x, y)
    if (initial == g)
    {
        System.out.print("POSSIBLE\n");
    }
    else
    {
        // Here it's never possible
        // if gcd is not same
        System.out.print("NOT POSSIBLE\n");
    }
}

// Driver code
public static void main(String args[])
{
    int A = 1, B = 2, X = 4, Y = 11;
    IsPossible(A, B, X, Y);

    A = 2; B = 2; X = 3; Y = 6;
    IsPossible(A, B, X, Y);
}
}

// This code is contributed by shivanisinghss2110

Python3

# Python3 program to find is it possible
# to have X white and Y black
# balls at the end.

# Recursive function to return
# gcd of a and b
def gcd(a, b) :

    if (b == 0) :
        return a;
    return gcd(b, a % b);


# Function returns if it's
# possible to have X white
# and Y black balls or not.
def IsPossible(a, b, x, y) :

    # Finding gcd of (x, y)
    # and (a, b)
    final = gcd(x, y);
    initial = gcd(a, b);

    # If gcd is same, it's always
    # possible to reach (x, y)
    if (initial == final) :
        print("POSSIBLE");

    else :

        # Here it's never possible
        # if gcd is not same
        print("NOT POSSIBLE");


# Driver Code
if __name__ == "__main__" :

    A = 1; B = 2; X = 4; Y = 11;
    IsPossible(A, B, X, Y);

    A = 2; B = 2; X = 3; Y = 6;
    IsPossible(A, B, X, Y);

# This code is contributed by AnkitRai01

C#

// C# program to Find is it possible
// to have X white and Y black
// balls at the end.
using System;
using System.Linq;

class GFG {

// Recursive function to return
// gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;

    return gcd(b, a % b);
}

// Function returns if it's
// possible to have X white
// and Y black balls or not.
static void IsPossible(int a, int b,
                       int x, int y)
{

    // Finding gcd of (x, y)
    // and (a, b)
    int g = gcd(x, y);
    int initial = gcd(a, b);

    // If gcd is same, it's always
    // possible to reach (x, y)
    if (initial == g)
    {
        Console.Write("POSSIBLE\n");
    }
    else
    {

        // Here it's never possible
        // if gcd is not same
        Console.Write("NOT POSSIBLE\n");
    }
}

// Driver code
static public void Main()
{
    int A = 1, B = 2;
    int X = 4, Y = 11;
    IsPossible(A, B, X, Y);

    A = 2; B = 2;
    X = 3; Y = 6;
    IsPossible(A, B, X, Y);
}
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
    // Javascript program to Find is it possible
    // to have X white and Y black
    // balls at the end.

    // Recursive function to return
    // gcd of a and b
    function gcd(a, b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }

    // Function returns if it's
    // possible to have X white
    // and Y black balls or not.
    function IsPossible(a, b, x, y)
    {

        // Finding gcd of (x, y)
        // and (a, b)
        let final = gcd(x, y);
        let initial = gcd(a, b);

         // If gcd is same, it's always
        // possible to reach (x, y)
        if (initial == final)
        {

            document.write("POSSIBLE" + "</br>");
        }
        else
        {
            // Here it's never possible
            // if gcd is not same
            document.write("NOT POSSIBLE" + "</br>");
        }
    }

    let A = 1, B = 2, X = 4, Y = 11;
    IsPossible(A, B, X, Y);

    A = 2, B = 2, X = 3, Y = 6;
    IsPossible(A, B, X, Y);

// This code is contributed by divyesh072019.
</script>

Output:

POSSIBLE
NOT POSSIBLE

Time Complexity: O(log(max(A, B, X, Y)))

Auxiliary Space: O(1)