# Coin game winner where every player has three choices

A and B are playing a game. At 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 coins. 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:

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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 ` `=` `5` `if` `(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

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