A and B are playing a game. In the beginning, there are n coins. Given two more numbers x and y. In each move, a player can pick x or y or 1 coin. A always starts the game. The player who picks the last coin wins the game or the person who is not able to pick any coin loses the game. For a given value of n, find whether A will win the game or not if both are playing optimally.
Examples:
Input : n = 5, x = 3, y = 4 Output : A There are 5 coins, every player can pick 1 or 3 or 4 coins on his/her turn. A can win by picking 3 coins in first chance. Now 2 coins will be left so B will pick one coin and now A can win by picking the last coin. Input : 2 3 4 Output : B
Let us take few example values of n for x = 3, y = 4.
n = 0 A can not pick any coin so he losses
n = 1 A can pick 1 coin and win the game
n = 2 A can pick only 1 coin. Now B will pick 1 coin and win the game
n = 3 4 A will win the game by picking 3 or 4 coins
n = 5, 6 A will choose 3 or 4 coins. Now B will have to choose from 2 coins so A will win.
We can observe that A wins game for n coins only when B loses for coins n-1 or n-x or n-y.
C++
// C++ program to find winner of game // if player can pick 1, x, y coins #include <bits/stdc++.h> using namespace std; // To find winner of game bool findWinner(int x, int y, int n) { // To store results int dp[n + 1]; // Initial values dp[0] = false; dp[1] = true; // Computing other values. for (int i = 2; i <= n; i++) { // If A losses any of i-1 or i-x // or i-y game then he will // definitely win game i if (i - 1 >= 0 and !dp[i - 1]) dp[i] = true; else if (i - x >= 0 and !dp[i - x]) dp[i] = true; else if (i - y >= 0 and !dp[i - y]) dp[i] = true; // Else A loses game. else dp[i] = false; } // If dp[n] is true then A will // game otherwise he losses return dp[n]; } // Driver program to test findWinner(); int main() { int x = 3, y = 4, n = 5; if (findWinner(x, y, n)) cout << 'A'; else cout << 'B'; return 0; } |
Java
// Java program to find winner of game // if player can pick 1, x, y coins import java.util.Arrays; public class GFG { // To find winner of game static boolean findWinner(int x, int y, int n) { // To store results boolean[] dp = new boolean[n + 1]; Arrays.fill(dp, false); // Initial values dp[0] = false; dp[1] = true; // Computing other values. for (int i = 2; i <= n; i++) { // If A losses any of i-1 or i-x // or i-y game then he will // definitely win game i if (i - 1 >= 0 && dp[i - 1] == false) dp[i] = true; else if (i - x >= 0 && dp[i - x] == false) dp[i] = true; else if (i - y >= 0 && dp[i - y] == false) dp[i] = true; // Else A loses game. else dp[i] = false; } // If dp[n] is true then A will // game otherwise he losses return dp[n]; } // Driver program to test findWinner(); public static void main(String args[]) { int x = 3, y = 4, n = 5; if (findWinner(x, y, n) == true) System.out.println('A'); else System.out.println('B'); } } // This code is contributed by Sumit Ghosh |
Python3
# Python3 program to find winner of game # if player can pick 1, x, y coins # To find winner of game def findWinner(x, y, n): # To store results dp = [0 for i in range(n + 1)] # Initial values dp[0] = False dp[1] = True # Computing other values. for i in range(2, n + 1): # If A losses any of i-1 or i-x # or i-y game then he will # definitely win game i if (i - 1 >= 0 and not dp[i - 1]): dp[i] = True elif (i - x >= 0 and not dp[i - x]): dp[i] = True elif (i - y >= 0 and not dp[i - y]): dp[i] = True # Else A loses game. else: dp[i] = False # If dp[n] is true then A will # game otherwise he losses return dp[n] # Driver Code x = 3; y = 4; n = 5if (findWinner(x, y, n)): print('A') else: print('B') # This code is contributed by Azkia Anam |
C#
// C# program to find winner of game // if player can pick 1, x, y coins using System; public class GFG { // To find winner of game static bool findWinner(int x, int y, int n) { // To store results bool[] dp = new bool[n + 1]; for(int i = 0; i < n+1; i++) dp[i] =false; // Initial values dp[0] = false; dp[1] = true; // Computing other values. for (int i = 2; i <= n; i++) { // If A losses any of i-1 or i-x // or i-y game then he will // definitely win game i if (i - 1 >= 0 && dp[i - 1] == false) dp[i] = true; else if (i - x >= 0 && dp[i - x] == false) dp[i] = true; else if (i - y >= 0 && dp[i - y] == false) dp[i] = true; // Else A loses game. else dp[i] = false; } // If dp[n] is true then A will // game otherwise he losses return dp[n]; } // Driver program to test findWinner(); public static void Main() { int x = 3, y = 4, n = 5; if (findWinner(x, y, n) == true) Console.WriteLine('A'); else Console.WriteLine('B'); } } // This code is contributed by vt_m. |
PHP
<?php // PHP program to find winner of game // if player can pick 1, x, y coins // To find winner of game function findWinner( $x, $y, $n) { // To store results $dp= array(); // Initial values $dp[0] = false; $dp[1] = true; // Computing other values. for ($i = 2; $i <= $n; $i++) { // If A losses any of i-1 or i-x // or i-y game then he will // definitely win game i if ($i - 1 >= 0 and !$dp[$i - 1]) $dp[$i] = true; else if ($i - $x >= 0 and !$dp[$i - $x]) $dp[$i] = true; else if ($i - $y >= 0 and !$dp[$i - $y]) $dp[$i] = true; // Else A loses game. else $dp[$i] = false; } // If dp[n] is true then A will // game otherwise he losses return $dp[$n]; } // Driver program to test findWinner(); $x = 3; $y = 4; $n = 5; if (findWinner($x, $y, $n)) echo 'A'; else echo 'B'; // This code is contributed by anuj_67. ?> |
Javascript
<script> // Javascript program to find winner of game // if player can pick 1, x, y coins // To find winner of game function findWinner(x, y, n) { // To store results var dp = Array(n + 1).fill(0); // Initial values dp[0] = false; dp[1] = true; // Computing other values. for(var i = 2; i <= n; i++) { // If A losses any of i-1 or i-x // or i-y game then he will // definitely win game i if (i - 1 >= 0 && !dp[i - 1]) dp[i] = true; else if (i - x >= 0 && !dp[i - x]) dp[i] = true; else if (i - y >= 0 && !dp[i - y]) dp[i] = true; // Else A loses game. else dp[i] = false; } // If dp[n] is true then A will // game otherwise he losses return dp[n]; } // Driver code var x = 3, y = 4, n = 5; if (findWinner(x, y, n)) document.write('A'); else document.write('B'); // This code is contributed by noob2000 </script> |
Output
A
Time Complexity: O(n)
Auxiliary Space: O(n)
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