Two players **player1** and **player2** are playing a game on a given number sequence **S** where **player1** starts first and both of them play optimally. The task is to find whether **player1** wins or loses. If he wins print “Yes” otherwise print “No”.

The rules of the game are as follows:

- The player’s alternate turns.
- In each turn, the current player must choose one or more elements of the current sequence S such that the values of all chosen elements are identical, and erase these elements from S.
- When a player cannot choose anything (the sequence S is already empty), this player loses the game.

**Examples:**

Input:S = {2, 1, 2, 2}

Output:Yes

Explanation:

The first player can choose the subset of elements with indices {0, 3} {0, 2} or {2, 3} (in the sequence) to ensure his victory in the future.

Input:S = {3, 2, 2, 3, 3, 5}

Output:No

Explanation:

The first player can not win in this sequence.

Ifplayer1chooses 2 and remove all the 2 from sequence thenplayer2will choose two 3 and remove only two 3 as both players play optimally.

Thenplayer1removes either 3 or 5 andplayer2will remove the final element. Soplayer1will always lose.

**Approach:** The approach to the above problem can be given as, if the total number of elements in sequence is even and the number of elements that occur more than once is also even then the first player can not win. If the number of elements that occur more than once is odd and the occurrence of repeated elements is greater than or equal to 4 and multiple of 2, then the first player can not win. Otherwise, **player1** can be the winner.

Below is the implementation of the above approach:

## C++

`// C++ implementation for Two player ` `// game in which a player can remove ` `// all occurrences of a number ` ` ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function that print whether ` `// player1 can wins or loses ` `void` `game(` `int` `v[], ` `int` `n) ` `{ ` ` ` `unordered_map<` `int` `, ` `int` `> m; ` ` ` ` ` `// storing the number of occurrence ` ` ` `// of elements in unordered map ` ` ` `for` `(` `int` `i = 0; i < n; i++) { ` ` ` ` ` `if` `(m.find(v[i]) == m.end()) ` ` ` `m[v[i]] = 1; ` ` ` ` ` `else` ` ` `m[v[i]]++; ` ` ` `} ` ` ` ` ` `int` `count = 0; ` ` ` ` ` `// variable to check if the ` `// occurrence of repeated ` `// elements is >= 4 and ` ` ` `// multiple of 2 or not ` ` ` `int` `check = 0; ` ` ` ` ` `// count elements which ` `// occur more than once ` ` ` `for` `(` `auto` `i: m) { ` ` ` `if` `(i.second > 1) { ` ` ` ` ` `if` `(i.second >= 4 ` `&& i.second % 2 == 0) ` ` ` `check++; ` ` ` ` ` `count++; ` ` ` `} ` ` ` `} ` ` ` ` ` `if` `(check % 2 != 0) ` ` ` `bool` `flag = ` `false` `; ` ` ` ` ` `if` `(check % 2 != 0) ` ` ` `cout << ` `"Yes"` `<< endl; ` ` ` ` ` `else` `if` `(n % 2 == 0 ` `&& count % 2 == 0) ` ` ` `cout << ` `"No"` `<< endl; ` ` ` ` ` `else` ` ` `cout << ` `"Yes"` `<< endl; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` ` ` `int` `arr[] = { 3, 2, 2, 3, 3, 5 }; ` ` ` ` ` `int` `size = ` `sizeof` `(arr) ` `/ ` `sizeof` `(arr[0]); ` ` ` ` ` `game(arr, size); ` ` ` ` ` `return` `0; ` `} ` |

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## Python3

`# Python3 implementation for two player ` `# game in which a player can remove ` `# all occurrences of a number ` `from` `collections ` `import` `defaultdict ` ` ` `# Function that print whether ` `# player1 can wins or loses ` `def` `game(v, n): ` ` ` ` ` `m ` `=` `defaultdict(` `int` `) ` ` ` ` ` `# Storing the number of occurrence ` ` ` `# of elements in unordered map ` ` ` `for` `i ` `in` `range` `(n): ` ` ` `if` `(v[i] ` `not` `in` `m): ` ` ` `m[v[i]] ` `=` `1` ` ` ` ` `else` `: ` ` ` `m[v[i]] ` `+` `=` `1` ` ` ` ` `count ` `=` `0` ` ` ` ` `# Variable to check if the ` ` ` `# occurrence of repeated ` ` ` `# elements is >= 4 and ` ` ` `# multiple of 2 or not ` ` ` `check ` `=` `0` ` ` ` ` `# Count elements which ` ` ` `# occur more than once ` ` ` `for` `i ` `in` `m.values(): ` ` ` `if` `(i > ` `1` `): ` ` ` `if` `(i >` `=` `4` `and` `i ` `%` `2` `=` `=` `0` `): ` ` ` `check ` `+` `=` `1` ` ` `count ` `+` `=` `1` ` ` ` ` `if` `(check ` `%` `2` `!` `=` `0` `): ` ` ` `flag ` `=` `False` ` ` ` ` `if` `(check ` `%` `2` `!` `=` `0` `): ` ` ` `print` `(` `"Yes"` `) ` ` ` ` ` `elif` `(n ` `%` `2` `=` `=` `0` `and` `count ` `%` `2` `=` `=` `0` `): ` ` ` `print` `(` `"No"` `) ` ` ` ` ` `else` `: ` ` ` `print` `(` `"Yes"` `) ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `arr ` `=` `[ ` `3` `, ` `2` `, ` `2` `, ` `3` `, ` `3` `, ` `5` `] ` ` ` `size ` `=` `len` `(arr) ` ` ` ` ` `game(arr, size) ` ` ` `# This code is contributed by chitranayal ` |

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**Output:**

No

**Time complexity:** The complexity of above approach is O(N).

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