# Two Balls Reachability Game

• Last Updated : 15 Nov, 2021

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)```
1.
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 ``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

 ``

Output:

```POSSIBLE
NOT POSSIBLE```

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

Auxiliary Space: O(1)

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