Largest odd divisor Game to check which player wins

Two players are playing a game starting with a number n. In each turn, a player can make any one of the subsequent moves: 
 

  • Divide n by any of its odd divisors greater than 1. Divisors of a number include the number itself.
  • Subtract 1 from n if n > k where k < n.

Player 1 makes the primary move, print “yes” if player 1 wins otherwise print “no” if both play optimally. The player who is unable to make a move loses the game.
Examples: 
 

Input: n = 12, k = 1 
Output: Yes 
Explanation: 
Player 1 first move = 12 / 3 = 4 
Player 2 first move = 4 – 1 = 3 
Player 1 second move = 3 / 3 = 1 
Player 2 second move can be done and hence he looses. 
Input: n = 1, k = 1
Output: No 
Explanation: 
Player 1 first move is not possible because n = k and hence player 1 looses.

 

Approach: The idea is to analyze the problem for the following 3 cases:
 



  • When integer n is odd, player 1 can divide n by itself, since it is odd and hence n / n = 1, and player 2 loses. Note that here n = 1 is an exception.
  • When integer n is even and has no odd divisors greater than 1 then n is of the form 2x. Player 1 is bound to subtract it by 1 making n odd. So if x > 1, player 2 wins. Note that for x = 1, n – 1 is equal to 1, so Player 1 wins.
  • When integer n is even and has odd divisors, the task remains to check if n is divisible by 4 then player 1 can divide n by its largest odd factor after which n becomes of the form 2x where x > 1, so again player 1 wins.
  • Otherwise, n must be of form 2 * p, where p is odd. If p is prime, player 1 loses since he can either reduce n by 1 or divide it by p both of which would be losing for him. If p is not prime then p must be of the form p1 * p2 where p1 is prime and p2 is any odd number > 1, for which player 1 can win by dividing n by p2.

Below is the implementation of the above approach:
 

C++

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// C++ implementation to find the
// Largest Odd Divisor Game to
// check which player wins
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the
// Largest Odd Divisor Game to
// check which player wins
void findWinner(int n, int k)
{
    int cnt = 0;
  
    // Check if n == 1 then
    // player 2 will win
    if (n == 1)
        cout << "No" << endl;
  
    // Check if n == 2 or n is odd
    else if ((n & 1) or n == 2)
        cout << "Yes" << endl;
  
    else {
        int tmp = n;
        int val = 1;
  
        // While n is greater than k and
        // divisible by 2 keep
        // incrementing tha val
        while (tmp > k and tmp % 2 == 0) {
            tmp /= 2;
            val *= 2;
        }
  
        // Loop to find greatest
        // odd divisor
        for (int i = 3; i <= sqrt(tmp); i++) {
            while (tmp % i == 0) {
                cnt++;
                tmp /= i;
            }
        }
        if (tmp > 1)
            cnt++;
  
        // Check if n is a power of 2
        if (val == n)
            cout << "No" << endl;
  
        else if (n / tmp == 2 and cnt == 1)
            cout << "No" << endl;
  
        // Check if cnt is not one
        // then player 1 wins
        else
            cout << "Yes" << endl;
    }
}
  
// Driver code
int main()
{
    long long n = 1, k = 1;
    findWinner(n, k);
    return 0;
}

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Java

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// Java implementation to find the
// Largest Odd Divisior Game to
// check which player wins
import java.util.*;
  
class GFG{
      
// Function to find the
// Largest Odd Divisior Game to
// check which player wins
public static void findWinner(int n, int k)
{
    int cnt = 0;
  
    // Check if n == 1 then
    // player 2 will win
    if (n == 1)
        System.out.println("No");
      
    // Check if n == 2 or n is odd
    else if ((n & 1) != 0 || n == 2)
        System.out.println("Yes");
  
    else
    {
        int tmp = n;
        int val = 1;
  
        // While n is greater than k and
        // divisible by 2 keep
        // incrementing tha val
        while (tmp > k && tmp % 2 == 0)
        {
            tmp /= 2;
            val *= 2;
        }
  
        // Loop to find greatest
        // odd divisor
        for(int i = 3
                i <= Math.sqrt(tmp); i++)
        {
           while (tmp % i == 0)
           {
               cnt++;
               tmp /= i;
           }
        }
        if (tmp > 1)
            cnt++;
  
        // Check if n is a power of 2
        if (val == n)
            System.out.println("No");
  
        else if (n / tmp == 2 && cnt == 1)
            System.out.println("No");
  
        // Check if cnt is not one
        // then player 1 wins
        else
            System.out.println("Yes");
    }
}
  
// Driver code
public static void main(String[] args)
{
    int n = 1, k = 1;
      
    findWinner(n, k);
}
}
  
// This code is contributed by jrishabh99 

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Python3

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# Python3 implementation to find  
# the Largest Odd Divisor Game 
# to check which player wins 
import math  
  
# Function to find the Largest 
# Odd Divisor Game to check
# which player wins 
def findWinner(n, k): 
      
    cnt = 0
  
    # Check if n == 1 then 
    # player 2 will win 
    if (n == 1):
        print("No"); 
  
    # Check if n == 2 or n is odd 
    elif ((n & 1) or n == 2):
        print("Yes"); 
  
    else:
        tmp = n; 
        val = 1
  
        # While n is greater than k and 
        # divisible by 2 keep 
        # incrementing tha val 
        while (tmp > k and tmp % 2 == 0): 
            tmp //= 2
            val *= 2
              
        # Loop to find greatest 
        # odd divisor 
        for i in range(3, int(math.sqrt(tmp)) + 1): 
            while (tmp % i == 0):
                cnt += 1
                tmp //= i; 
          
        if (tmp > 1):
            cnt += 1
  
        # Check if n is a power of 2 
        if (val == n):
            print("No"); 
  
        elif (n / tmp == 2 and cnt == 1):
            print("No"); 
  
        # Check if cnt is not one 
        # then player 1 wins 
        else:
            print("Yes"); 
              
# Driver code 
if __name__ == "__main__"
  
    n = 1; k = 1
      
    findWinner(n, k); 
  
# This code is contributed by AnkitRai01

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C#

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// C# implementation to find the
// Largest Odd Divisior Game to
// check which player wins
using System;
   
class GFG{
       
// Function to find the
// Largest Odd Divisior Game to
// check which player wins
public static void findWinner(int n, int k)
{
    int cnt = 0;
   
    // Check if n == 1 then
    // player 2 will win
    if (n == 1)
        Console.Write("No");
       
    // Check if n == 2 or n is odd
    else if ((n & 1) != 0 || n == 2)
        Console.Write("Yes");
   
    else
    {
        int tmp = n;
        int val = 1;
   
        // While n is greater than k and
        // divisible by 2 keep
        // incrementing tha val
        while (tmp > k && tmp % 2 == 0)
        {
            tmp /= 2;
            val *= 2;
        }
   
        // Loop to find greatest
        // odd divisor
        for(int i = 3; 
                i <= Math.Sqrt(tmp); i++)
        {
           while (tmp % i == 0)
           {
               cnt++;
               tmp /= i;
           }
        }
        if (tmp > 1)
            cnt++;
   
        // Check if n is a power of 2
        if (val == n)
            Console.Write("No");
   
        else if (n / tmp == 2 && cnt == 1)
            Console.Write("No");
   
        // Check if cnt is not one
        // then player 1 wins
        else
            Console.Write("Yes");
    }
}
   
// Driver code
public static void Main(string[] args)
{
    int n = 1, k = 1;
       
    findWinner(n, k);
}
}
// This code is contributed by rutvik_56

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Output:

No

Time Complexity: O(sqrt(n)) 
Auxiliary Space: O(1)
 

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