Given an array **arr[]** of length, **N**, the task is to find the winner of a game played by two players **A** and **B** optimally, by performing the following operations:

- Player
**A**makes the first move. - Players need to alternately place
**+**and**–**sign in front of array elements in their turns. - After having placed signs in front of all array elements, player
**A**wins if the difference of all the elements is even. - Otherwise, player
**B**wins.

**Examples:**

Input:arr[] = {1, 2}Output:BExplanation:

All possible ways the game can be played out are:

(+1) – (+2) = -1

(−1) – (+2) = -3

(+1) – (-2) = 3

(-1) – (-2) = 1

Since the differences are odd in all the possibilities, B wins.

Input:arr[] = {1, 1, 2}Output:AExplanation:

All possible ways the game can be played out are:

(1) – (1) – (2) = -2

(1) – (1) – (-2) = 2

(1) – (-1) – (2) = 0

(1) – (-1) – (-2) = 4

(-1) – (1) – (2) = -4

(-1) – (1) – (-2) = 0

(-1) – (-1) – (2) = -2

(-1) – (-1) – (-2) = 4

Since the differences are even in all the possibilities, A wins.

**Naive Approach: **The simplest approach is to generate all the possible 2^{N} combinations in which the signs can be placed in the array and check for each combination, check if player A can win or not. If found to be true for any permutation, then print **A.** Otherwise, player **B** wins.

**Time Complexity:** O(2^{N} * N)**Auxiliary Space: **O(1)

**Efficient Approach:** Follow the steps below to optimize the above approach:

- Initialize a variable, say
**diff**, to store the sum of the array elements. - Traverse the array
**arr[]**, over the range of indices**[1, N]**, and update**diff**by subtracting**arr[i]**to it. - If
**diff % 2**is found to be equal to**0**, then print**‘A’**. Otherwise, print**‘B’**.

Below is the implementation of the above approach:

## C++

`// C++ program for the` `// above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to check which` `// player wins the game` `void` `checkWinner(` `int` `arr[], ` `int` `N)` `{` ` ` `// Stores the difference between` ` ` `// +ve and -ve array elements` ` ` `int` `diff = 0;` ` ` `// Traverse the array` ` ` `for` `(` `int` `i = 0; i < N; i++) {` ` ` `// Update diff` ` ` `diff -= arr[i];` ` ` `}` ` ` `// Checks if diff is even` ` ` `if` `(diff % 2 == 0) {` ` ` `cout << ` `"A"` `;` ` ` `}` ` ` `else` `{` ` ` `cout << ` `"B"` `;` ` ` `}` `}` `// Driver Code` `int` `main()` `{` ` ` `// Given Input` ` ` `int` `arr[] = { 1, 2 };` ` ` `int` `N = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `// Function call to check` ` ` `// which player wins the game` ` ` `checkWinner(arr, N);` ` ` `return` `0;` `}` |

## Python3

`# Python3 program for the` `# above approach` `# Function to check which` `# player wins the game` `def` `checkWinner(arr, N):` ` ` `# Stores the difference between` ` ` `# +ve and -ve array elements` ` ` `diff ` `=` `0` ` ` `# Traverse the array` ` ` `for` `i ` `in` `range` `(N):` ` ` `# Update diff` ` ` `diff ` `-` `=` `arr[i]` ` ` `# Checks if diff is even` ` ` `if` `(diff ` `%` `2` `=` `=` `0` `):` ` ` `print` `(` `"A"` `)` ` ` `else` `:` ` ` `print` `(` `"B"` `)` `# Driver Code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `# Given Input` ` ` `arr ` `=` `[` `1` `, ` `2` `]` ` ` `N ` `=` `len` `(arr)` ` ` `# Function call to check` ` ` `# which player wins the game` ` ` `checkWinner(arr, N)` ` ` `# This code is contributed by ukasp.` |

## Java

`// Java program for the` `// above approach` `import` `java.util.*;` `class` `GFG` `{` `// Function to check which` `// player wins the game` `static` `void` `checkWinner(` `int` `arr[], ` `int` `N)` `{` ` ` `// Stores the difference between` ` ` `// +ve and -ve array elements` ` ` `int` `diff = ` `0` `;` ` ` `// Traverse the array` ` ` `for` `(` `int` `i = ` `0` `; i < N; i++) {` ` ` `// Update diff` ` ` `diff -= arr[i];` ` ` `}` ` ` `// Checks if diff is even` ` ` `if` `(diff % ` `2` `== ` `0` `) {` ` ` `System.out.println(` `"A"` `);` ` ` `}` ` ` `else` `{` ` ` `System.out.println(` `"B"` `);` ` ` `}` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `// Given Input` ` ` `int` `arr[] = { ` `1` `, ` `2` `};` ` ` `int` `N = arr.length;` ` ` `// Function call to check` ` ` `// which player wins the game` ` ` `checkWinner(arr, N);` `}` `}` `// This code is contributed by Stream-Cipher` |

## C#

`// C# program for the above approach` `using` `System;` `class` `GFG{` `// Function to check which` `// player wins the game` `static` `void` `checkWinner(` `int` `[] arr, ` `int` `N)` `{` ` ` ` ` `// Stores the difference between` ` ` `// +ve and -ve array elements` ` ` `int` `diff = 0;` ` ` `// Traverse the array` ` ` `for` `(` `int` `i = 0; i < N; i++)` ` ` `{` ` ` ` ` `// Update diff` ` ` `diff -= arr[i];` ` ` `}` ` ` `// Checks if diff is even` ` ` `if` `(diff % 2 == 0)` ` ` `{` ` ` `Console.Write(` `"A"` `);` ` ` `}` ` ` `else` ` ` `{` ` ` `Console.Write(` `"B"` `);` ` ` `}` `}` `// Driver Code` `public` `static` `void` `Main()` `{` ` ` ` ` `// Given Input` ` ` `int` `[] arr = { 1, 2 };` ` ` `int` `N = arr.Length;` ` ` ` ` `// Function call to check` ` ` `// which player wins the game` ` ` `checkWinner(arr, N);` `}` `}` `// This code is contributed by sanjoy_62` |

## Javascript

`<script>` `// JavaScript program for the above approach` `// Function to check which` `// player wins the game` `function` `checkWinner(arr, N)` `{` ` ` `// Stores the difference between` ` ` `// +ve and -ve array elements` ` ` `let diff = 0;` ` ` ` ` `// Traverse the array` ` ` `for` `(let i = 0; i < N; i++) {` ` ` `// Update diff` ` ` `diff -= arr[i];` ` ` `}` ` ` ` ` `// Checks if diff is even` ` ` `if` `(diff % 2 == 0) {` ` ` `document.write(` `"A"` `);` ` ` `}` ` ` `else` `{` ` ` `document.write(` `"B"` `);` ` ` `}` `}` `// Driver Code` ` ` `// Given Input` ` ` `let arr = [ 1, 2 ];` ` ` `let N = arr.length;` ` ` `// Function call to check` ` ` `// which player wins the game` ` ` `checkWinner(arr, N);` ` ` `</script>` |

**Output:**

B

**Time Complexity:** O(N)**Auxiliary Space:** O(1)

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