Optimal Strategy for the Divisor game using Dynamic Programming

Last Updated : 10 Mar, 2023

Given an integer N and two players, A and B are playing a game. On each player’s turn, that player makes a move by subtracting a divisor of current N (which is less than N) from current N, thus forming a new N for the next turn. The player who does not have any divisor left to subtract loses the game. The task is to tell which player wins the game if player A takes the first turn, assuming both players play optimally.
Examples:

Input : N = 2
Output : Player A wins
Explanation :
Player A chooses 1, and B has no more moves.
Input : N = 3
Output : Player B wins
Explanation :
Player A chooses 1, player B chooses 1, and A has no more moves.

Approach :
This problem mentioned above can be solved using Dynamic Programming.

We will take a DP having 2 states i.e.

N -> current number left
A -> boolean value to decide if it's player A's turn or not

At each state, we will try to find all the divisors of N and will try to find the next state where the current player is winning. For player A, we will try to find the next state where the return value is true while for player B, we will try to find the next state where the return value is false (as false represents the loss of player A).
The base cases will be for N=1 where always the player A will lose and N=2 where always the player B will lose.
To find the answer, we just need to find the value of DP[ N ][ 1 ].

Below is the implementation of the above approach:

C++

// C++ program for implementation of // Optimal Strategy for the Divisor // Game using Dynamic Programming #include <bits/stdc++.h> using namespace std;  // Recursive function to find the winner bool divisorGame(int N, bool A, int dp[][2]) {      // check if N=1 or N=3 then player B wins     if (N == 1 or N == 3)         return false;      // check if N=2 then player A wins     if (N == 2)         return true;      // check if current state already visited     // then return the previously obtained ans     if (dp[N][A] != -1)         return dp[N][A];      // check if currently it is player A's turn     // then initialise the ans to 0     int ans = (A == 1) ? 0 : 1;      // Traversing across all the divisors of N     // which are less than N     for (int i = 1; i * i <= N; i++) {          // check if current value of         // i is a divisor of N         if (N % i == 0) {              // check if it is player A's turn             // then we need at least one true             if (A)                 ans |= divisorGame(N - i, 0, dp);              // Else if it is player B's turn             // then we need at least one false             else                 ans &= divisorGame(N - i, 1, dp);         }     }      // Return the current ans     return dp[N][A] = ans; }  // Driver code int main() {     // initialise N     int N = 3;      int dp[N + 1][2];      memset(dp, -1, sizeof(dp));      if (divisorGame(N, 1, dp) == true)         cout << "Player A wins";     else         cout << "Player B wins";      return 0; }

Java

// Java program for implementation of // optimal strategy for the divisor // game using dynamic programming  import java.util.*;  class GFG {      // Recursive function to find the winner     static int divisorGame(int N, int A, int dp[][]) {          // Check if N = 1 or N = 3 then player B wins         if (N == 1 || N == 3)             return 0;          // Check if N = 2 then player A wins         if (N == 2)             return 1;          // Check if current state already visited         // then return the previously obtained ans         if (dp[N][A] != -1)             return dp[N][A];          // Check if currently it is player A's turn         // then initialise the ans to 0         int ans = (A == 1) ? 0 : 1;          // Traversing across all the divisors of N         // which are less than N         for (int i = 1; i * i <= N; i++) {              // Check if current value of             // i is a divisor of N             if (N % i == 0) {                  // Check if it is player A's turn                 // then we need at least one true                 if (A == 1)                     ans |= divisorGame(N - i, 0, dp);                  // Else if it is player B's turn                 // then we need at least one false                 else                    ans &= divisorGame(N - i, 1, dp);             }         }          // Return the current ans         return dp[N][A] = ans;     }      // Driver code     public static void main(String[] args) {                  // Initialise N         int N = 3;          int[][] dp = new int[N + 1][2];          for (int i = 0; i < N + 1; i++) {              for (int j = 0; j < 2; j++) {                   dp[i][j] = -1;             }         }          if (divisorGame(N, 1, dp) == 1)             System.out.print("Player A wins");         else             System.out.print("Player B wins");      } }  // This code contributed by sapnasingh4991

Python3

# Python3 program for implementation of # Optimal Strategy for the Divisor # Game using Dynamic Programming  from math import sqrt  # Recursive function to find the winner def divisorGame(N,A,dp):     # check if N=1 or N=3 then player B wins     if (N == 1 or N == 3):         return False      # check if N=2 then player A wins     if (N == 2):         return True      # check if current state already visited     # then return the previously obtained ans     if (dp[N][A] != -1):         return dp[N][A]      # check if currently it is player A's turn     # then initialise the ans to 0     if(A == 1):         ans = 0     else:         ans = 1      # Traversing across all the divisors of N     # which are less than N     for i in range(1,int(sqrt(N))+1,1):         # check if current value of         # i is a divisor of N         if (N % i == 0):             # check if it is player A's turn             # then we need at least one true             if (A):                 ans |= divisorGame(N - i, 0, dp)              # Else if it is player B's turn             # then we need at least one false             else:                 ans &= divisorGame(N - i, 1, dp)       dp[N][A] = ans       # Return the current ans     return dp[N][A]  # Driver code if __name__ == '__main__':     # initialise N     N = 3      dp = [[-1 for i in range(2)] for j in range(N+1)]      if (divisorGame(N, 1, dp) == True):         print("Player A wins")     else:         print("Player B wins")  # This code is contributed by Surendra_Gangwar

// C# program for implementation of // optimal strategy for the divisor // game using dynamic programming using System;  class GFG {  // Recursive function to find the winner static int divisorGame(int N, int A,                         int [,]dp) {      // Check if N = 1 or N = 3      // then player B wins     if (N == 1 || N == 3)         return 0;      // Check if N = 2 then player A wins     if (N == 2)         return 1;      // Check if current state already visited     // then return the previously obtained ans     if (dp[N, A] != -1)         return dp[N, A];      // Check if currently it is player A's turn     // then initialise the ans to 0     int ans = (A == 1) ? 0 : 1;      // Traversing across all the divisors of N     // which are less than N     for(int i = 1; i * i <= N; i++)     {                 // Check if current value of        // i is a divisor of N        if (N % i == 0)        {                        // Check if it is player A's turn            // then we need at least one true            if (A == 1)                ans |= divisorGame(N - i, 0, dp);                            // Else if it is player B's turn            // then we need at least one false            else               ans &= divisorGame(N - i, 1, dp);        }     }          // Return the current ans     return dp[N, A] = ans; }  // Driver code public static void Main(String[] args) {          // Initialise N     int N = 3;     int[,] dp = new int[N + 1, 2];          for(int i = 0; i < N + 1; i++)      {        for(int j = 0; j < 2; j++)         {           dp[i, j] = -1;        }     }          if (divisorGame(N, 1, dp) == 1)     {         Console.Write("Player A wins");     }     else     {         Console.Write("Player B wins");     } } }  // This code is contributed by amal kumar choubey

JavaScript

<script> // Javascript program for implementation of // Optimal Strategy for the Divisor // Game using Dynamic Programming   // Recursive function to find the winner function divisorGame(N, A, dp) {      // check if N=1 or N=3 then player B wins     if (N == 1 || N == 3)         return false;      // check if N=2 then player A wins     if (N == 2)         return true;      // check if current state already visited     // then return the previously obtained ans     if (dp[N][A] != -1)         return dp[N][A];      // check if currently it is player A's turn     // then initialise the ans to 0     let ans = (A == 1) ? 0 : 1;      // Traversing across all the divisors of N     // which are less than N     for (let i = 1; i * i <= N; i++) {          // check if current value of         // i is a divisor of N         if (N % i == 0) {              // check if it is player A's turn             // then we need at least one true             if (A)                 ans |= divisorGame(N - i, 0, dp);              // Else if it is player B's turn             // then we need at least one false             else                 ans &= divisorGame(N - i, 1, dp);         }     }      // Return the current ans     return dp[N][A] = ans; }  // Driver code  // initialise N let N = 3;  let dp = [];  for (let i = 0; i < N + 1; i++) {     let temp = [-1]     for (let j = 0; j < 2; j++) {         temp.push([-1])     }     dp.push(temp) }  // memset(dp, -1, sizeof(dp));  if (divisorGame(N, 1, dp) == true)     document.write("Player A wins"); else     document.write("Player B wins");  // This code is contributed by gfgking </script>

Output:

Player B wins

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

Game of N stones where each player can remove 1, 3 or 4

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Optimal Strategy for the Divisor game using Dynamic Programming

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