There are two players A and B and a pile of N cards stacked upon each other. The task is to find the winner of the game, assuming both players play optimally as per the following guidelines:
- Player A always begins the game and the players take alternate turns subsequently.
- In each turn, a player can remove K( 1 ? K ? N) cards if K & n = 0, where n is the size of the current pile.
- If a player cannot make a move at any point in the game, then that player loses, and the game ends.
Examples:
Input: N = 1
Output: B
Explanation:
A can only remove 1 card, but 1 & 1 = 1, so A is unable to make a move.
Hence, B wins the game.Input: N = 4
Output: A
Explanation:
A will remove 3 cards as 3 & 4 = 0, now only 1 card is left and B cannot make a move.
Hence, A wins the game.
Approach: The idea is based on the observation that if the count of 1s in the binary representation of N, before a 0 is encountered, is odd then A wins the game. If no such combination of 1s and 0s exists throughout the binary string, then B wins. Follow the steps below to solve the problem:
- Initialize a variable countOne to store the count of 1.
- Convert N into its binary representation and store it in a string, binString.
- Traverse the string binString and do the following:
- If '1' is encountered, increment the countOne.
- If '0' is encountered, check if the countOne is odd or even, if the countOne is odd A wins, and break out of the loop, otherwise reset the countOne to 0 and continue traversing.
- If the whole string is traversed without breaking, then B wins.
Below is the implementation of the above approach:
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to find the winner of the
// game if both player plays optimally
void findWinner(int N)
{
// Stores the count of 1s
int onesBeforeZero = 0;
int flag = 1;
// Convert N to binary representation
int binString[32];
int i = 0;
while (N > 0)
{
// Storing remainder in binary array
binString[i] = N % 2;
N = N / 2;
i++;
}
int l = sizeof(binString) /
sizeof(binString[0]);
// Traverse the binary string
for(int j = 0; j < l; j++)
{
// If 1 is encountered,
// increment count of 1s
if (binString[j] == 1)
{
onesBeforeZero += 1;
}
// If 0 is encountered, check
// if count of 1s is odd
else
{
// If count of 1s is odd,
// then winner is A
if (onesBeforeZero & 1)
{
cout << "A";
flag = 0;
break;
}
// If count of 1s is even,
// reset it to 0
else
onesBeforeZero = 0;
}
}
// If entire loop is traversed
// without breaking, then
// B is the winner
if (flag == 1)
cout << "B";
}
// Driver Code
int main()
{
int N = 4;
// Function Call
findWinner(N);
return 0;
}
// This code is contributed by jana_sayantan
// C program for the above approach
#include <stdio.h>
// Function to find the winner of the
// game if both player plays optimally
void findWinner(unsigned long long N)
{
// Stores the count of 1s
int onesBeforeZero = 0;
int flag = 1, j = 0;
char binString[32];
// Converting N into a binary string
for(int i = 31; i >= 0; i--)
{
unsigned long long temp = N >> i;
if (temp & 1)
binString[j] = '1';
else
binString[j] = '0';
j += 1;
}
// Traverse the binary string
for(int i = 0; i < 32; i++)
{
if (binString[i] == '1')
// If 1 is encountered
// increment ones count
onesBeforeZero += 1;
else
{
// If 0 is encountered check
// if ones count is odd
if (onesBeforeZero & 1)
{
// If ones count is odd
// winner is A break
printf("A");
flag = 0;
break;
}
else
// If ones count is even
// reset it to 0 and continue
onesBeforeZero = 0;
}
}
// If entire loop is traversed
// without breaking, then
// B is the winner
if (flag == 1)
printf("B");
}
// Driver code
int main()
{
unsigned long long N = 4;
// Function Call
findWinner(N);
return 0;
}
// This code is contributed by Praneeth Kapila
// Java program for the above approach
class GFG{
// Function to find the winner
static void findWinner(long N)
{
// Stores the count of 1s
int onesBeforeZero = 0, flag = 1, j = 0;
String[] binString = new String[32];
// Converting N into a binary string
for(int i = 31; i >= 0; i--)
{
long temp = N >> i;
if ((temp & 1) == 1)
binString[j] = "1";
else
binString[j] = "0";
j += 1;
}
for(int i = 0; i < 32; i++)
{
if (binString[i] == "1")
// If 1 is encountered
// increment ones count
onesBeforeZero += 1;
else
{
// If 0 is encountered check
//if ones count is odd
if ((onesBeforeZero & 1) == 1)
{
// If ones count is odd winner
// is A break
System.out.println("A");
flag = 0;
break;
}
else
// If ones count is even
// reset it to 0 and continue
onesBeforeZero = 0;
}
}
// If entire loop is traversed
// without breaking, then
// B is the winner
if (flag == 1)
System.out.println("B");
}
// Driver code
public static void main(String[] args)
{
long N = 4;
// Function Call
findWinner(N);
}
}
// This code is contributed by Praneeth Kapila
# Python3 program for the above approach
# Function to find the winner of the
# game if both player plays optimally
def findWinner(N):
# Stores the count of 1s
onesBeforeZero = 0
flag = 1
# Convert N to binary representation
binString = bin(N).replace("0b", "")
l = len(binString)
# Traverse the binary string
for j in range(l):
# If 1 is encountered,
# increment count of 1s
if binString[j] == '1':
onesBeforeZero += 1
# If 0 is encountered, check
# if count of 1s is odd
else:
# If count of 1s is odd,
# then winner is A
if onesBeforeZero & 1:
print("A")
flag = 0
break
# If count of 1s is even,
# reset it to 0
else:
onesBeforeZero = 0
# If entire loop is traversed
# without breaking, then
# B is the winner
if flag == 1:
print("B")
# Driver Code
N = 4
# Function Call
findWinner(N)
// C# program for the above approach
using System;
class GFG{
// Function to find the winner
static void findWinner(long N)
{
// Stores the count of 1s
int onesBeforeZero = 0, flag = 1, j = 0;
String[] binString = new String[32];
// Converting N into a binary string
for(int i = 31; i >= 0; i--)
{
long temp = N >> i;
if ((temp & 1) == 1)
binString[j] = "1";
else
binString[j] = "0";
j += 1;
}
for(int i = 0; i < 32; i++)
{
if (binString[i] == "1")
// If 1 is encountered
// increment ones count
onesBeforeZero += 1;
else
{
// If 0 is encountered check
//if ones count is odd
if ((onesBeforeZero & 1) == 1)
{
// If ones count is odd winner
// is A break
Console.WriteLine("A");
flag = 0;
break;
}
else
// If ones count is even
// reset it to 0 and continue
onesBeforeZero = 0;
}
}
// If entire loop is traversed
// without breaking, then
// B is the winner
if (flag == 1)
Console.WriteLine("B");
}
// Driver code
public static void Main(String[] args)
{
long N = 4;
// Function Call
findWinner(N);
}
}
// This code is contributed by shivanisinghss2110
<script>
// Javascript program for the above approach
// Function to find the winner
function findWinner(N)
{
// Stores the count of 1s
let onesBeforeZero = 0, flag = 1, j = 0;
let binString = [];
// Converting N into a binary string
for(let i = 31; i >= 0; i--)
{
let temp = N >> i;
if ((temp & 1) == 1)
binString[j] = "1";
else
binString[j] = "0";
j += 1;
}
for(let i = 0; i < 32; i++)
{
if (binString[i] == "1")
// If 1 is encountered
// increment ones count
onesBeforeZero += 1;
else
{
// If 0 is encountered check
//if ones count is odd
if ((onesBeforeZero & 1) == 1)
{
// If ones count is odd winner
// is A break
document.write("A");
flag = 0;
break;
}
else
// If ones count is even
// reset it to 0 and continue
onesBeforeZero = 0;
}
}
// If entire loop is traversed
// without breaking, then
// B is the winner
if (flag == 1)
document.write("B");
}
// Driver code
let N = 4;
// Function Call
findWinner(N);
// This code is contributed by code_hunt
</script>
Output:
A
Time Complexity: O(log N)
Auxiliary Space: O(log N)