Find winner of the game when any set bit is removed in each move

Two players, Player 1 and Player 2, are given an integer N to play a game. The rules of the game are as follows : 

• In one turn, a player can remove any set bit of N in its binary representation to make a new N. 
• Player 1 always takes the first turn. 
• If a player cannot make a move, he loses.

Examples:

Input: N = 8
Output: 1
Explanation: N = 8
N = 1000 (binary)
Player 1 takes the bit.
The remaining bits are all zero.
Player 2 cannot make a move,  
so Player 1 wins.

Input: N = 3
Output: 2

Approach: The given problem can be solved by following the below idea:

Calculate the number of set bits in N. If the number of set bits is odd then player 1 will always win [because he will take the following turns – 1^st, 3^rd, 5^th, . . . and any odd turn]. Otherwise, player 2 will win the game.

Follow the steps mentioned below to implement the:

• Calculate the number of set bits in N.
• If the number of the set bits is odd then player 1 wins.
• Else, player 2 wins.

Below is the implementation of the above approach.

C++

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the winner
int swapBitGame(long long N)
{
    int bitCount = 0;

    // Calculate the number of set bit in
    // N using Brian Kernighan's Algorithm
    while (N) {
        N = (N & (N - 1));
        bitCount++;
    }

    // If bitCount is even return 2
    // else return 1
    return bitCount % 2 == 0 ? 2 : 1;
}

// Driver Code
int main()
{
    long long N = 8;

    // Function Call
    cout << swapBitGame(N) << endl;

    return 0;
}

Java

// Java code to implement the approach
import java.io.*;

class GFG {

  // Function to find the winner
  static int swapBitGame(long N)
  {
    int bitCount = 0;

    // Calculate the number of set bit in
    // N using Brian Kernighan's Algorithm
    while (N!=0) {
      N = (N & (N - 1));
      bitCount++;
    }

    // If bitCount is even return 2
    // else return 1
    return bitCount % 2 == 0 ? 2 : 1;
  }

  public static void main(String[] args)
  {
    long N = 8;

    // Function call
    System.out.println(swapBitGame(N));
  }
}

// This code is contributed by lokeshmvs21.

Python

# Python code to implement the approach

# Function to find the winner
def swapBitGame(N):

    bitCount = 0

    # Calculate the number of set bit in
    # N using Brian Kernighan's Algorithm
    while (N > 0):
        N = (N & (N - 1))
        bitCount += 1

    # If bitCount is even return 2
    # else return 1
    return (2) if bitCount % 2 == 0 else (1)

# Driver Code
N = 8

#Function Call
print(swapBitGame(N))

# This code is contributed by Samim Hossain Mondal.

C#

// C# code to implement the approach
using System;

public class GFG {

  // Function to find the winner
  static int swapBitGame(long N)
  {
    int bitCount = 0;

    // Calculate the number of set bit in
    // N using Brian Kernighan's Algorithm
    while (N != 0) {
      N = (N & (N - 1));
      bitCount++;
    }

    // If bitCount is even return 2
    // else return 1
    return bitCount % 2 == 0 ? 2 : 1;
  }

  static public void Main()
  {

    // Code
    long N = 8;

    // Function call
    Console.WriteLine(swapBitGame(N));
  }
}

// This code is contributed by lokesh.

Javascript

// JavaScript code to implement the approach

// Function to find the winner
const swapBitGame = (N) => {
    let bitCount = 0;

    // Calculate the number of set bit in
    // N using Brian Kernighan's Algorithm
    while (N) {
        N = (N & (N - 1));
        bitCount++;
    }

    // If bitCount is even return 2
    // else return 1
    return bitCount % 2 == 0 ? 2 : 1;
}

// Driver Code
let N = 8;

// Function Call
console.log(swapBitGame(N));

// This code is contributed by rakeshsahni.

Output

1

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

Approach: Naive Recursion with memoization

Steps:

• First, check if the solution for the given input N is already present in the memoization map.
• If it is present, then return the value of the solution from the map to avoid recalculation.
• If it is not present, use a loop to iterate through all the bits of N.
• Remove the set bit by using the XOR operation.
• Then recursively call the solve function for the new value of N.
• After the recursive call, we use the OR operation to set the bit back to the original value of N.
• Check if the returned solution from the recursive call is Player 2 losing or not. If it is Player 2 losing.
• update the memoization map with the solution and return Player 1 wins. 
• If we have iterated through all the bits of N and Player 2 has not lost.
• update the memoization map with Player 2 winning and return Player 2 wins.
• Finally, print the result.

Below is the code implementation of the above approach:

C++

// C++ program for the above problem using Naive Recursion with Memoization approach
#include<bits/stdc++.h>
using namespace std;

// Memoization table
unordered_map<int, int> memo;

// Recursive function
int solve(int n){
      // Base case
    if(n==0) return 2;
    if(memo.find(n) != memo.end()) return memo[n];
    int ans = 0;
    for(int i=0; i<31; i++){
        if(n & (1<<i)){
            n ^= (1<<i);
            ans = solve(n);
            n |= (1<<i);
              // If Player 1 wins in the current subproblem, then Player 2 will lose, so return 1.
            if(ans==2) {
                memo[n] = 1;
                return 1;
            }
        }
    }
      // If Player 1 cannot make any move, then Player 2 wins.
    memo[n] = 2;
    return 2;
}

// Driver code
int main(){
    int n=8;
    int res = solve(n);
    if(res==1) cout << "1";
    else cout << "2";
    return 0;
}

Java

// Java code

import java.io.*;
import java.util.HashMap;
import java.util.Map;

public class GFG {
    // Memoization table
    static Map<Integer, Integer> memo = new HashMap<>();

    // Recursive function
    static int solve(int n) {
        // Base case
        if (n == 0) return 2;
        if (memo.containsKey(n)) return memo.get(n);
        int ans = 0;
        for (int i = 0; i < 31; i++) {
            if ((n & (1 << i)) != 0) {
                n ^= (1 << i);
                ans = solve(n);
                n |= (1 << i);
                // If Player 1 wins in the current subproblem,
                // then Player 2 will lose, so return 1.
                if (ans == 2) {
                    memo.put(n, 1);
                    return 1;
                }
            }
        }
        // If Player 1 cannot make any move, then Player 2 wins.
        memo.put(n, 2);
        return 2;
    }

    // Driver code
    public static void main(String[] args) {
        int n = 8;
        int res = solve(n);
        if (res == 1) {
            System.out.println("1");
        } else {
            System.out.println("2");
        }
    }
}

// This code is contributed by guptapratik

Python3

import sys

#  Memoization table
memo = {}

# Recursive function
def solve(n):

  # Base case
    if n == 0:
        return 2
    if n in memo:
        return memo[n]
    ans = 0
    for i in range(31):
        if n & (1 << i):
            n ^= (1 << i)
            ans = solve(n)
            n |= (1 << i)
            # If Player 1 wins in the current subproblem,
            # then Player 2 will lose, so return 1.
            if ans == 2:
                memo[n] = 1
                return 1
    # If Player 1 cannot make any move, then Player 2 wins.
    memo[n] = 2
    return 2

# Test case
n = 8
res = solve(n)
if res == 1:
    print("1")
else:
    print("2")
sys.exit(0)

C#

using System;
using System.Collections.Generic;

class Program
{
    // Memoization table
    static Dictionary<int, int> memo = new Dictionary<int, int>();

    // Recursive function
    static int Solve(int n)
    {
        // Base case
        if (n == 0) return 2;
        if (memo.ContainsKey(n)) return memo[n];
        int ans = 0;

        for (int i = 0; i < 31; i++)
        {
            if ((n & (1 << i)) != 0)
            {
                n ^= (1 << i);
                ans = Solve(n);

                // If Player 1 wins in the current subproblem, then Player 2 will lose, so return 1.
                if (ans == 2)
                {
                    memo[n] = 1;
                    return 1;
                }

                n |= (1 << i);
            }
        }

        // If Player 1 cannot make any move, then Player 2 wins.
        memo[n] = 2;
        return 2;
    }

    // Driver code
    static void Main()
    {
        int n = 8;
        int res = Solve(n);
        if (res == 1) Console.WriteLine("1");
        else Console.WriteLine("2");
    }
}

Javascript

// Memoization table
const memo = new Map();

// Recursive function
function solve(n) {
    // Base case
    if (n === 0) return 2;
    if (memo.has(n)) return memo.get(n);
    let ans = 0;
    for (let i = 0; i < 31; i++) {
        if (n & (1 << i)) {
            n ^= (1 << i);
            ans = solve(n);
            n |= (1 << i);

             // If Player 1 wins in the current subproblem,
             // then Player 2 will lose, so return 1.
            if (ans === 2) {
                memo.set(n, 1);
                return 1;
            }
        }
    }
     // If Player 1 cannot make any move, then Player 2 wins.
    memo.set(n, 2);
    return 2;
}

// Driver code
const n = 8;
const res = solve(n);
if (res === 1) console.log("1");
else console.log("2");

Output

1

Time Complexity: O(logN)
Auxiliary Space: O(N)