Rat and Poisoned bottle Problem

Given N number of bottles in which one bottle is poisoned. So the task is to find out minimum number of rats required to identify the poisoned bottle. A rat can drink any number of bottles at a time.

Examples:

Input: N = 4 
Output: 2

Input: N = 100
Output: 7

Approach:
Let’s start from the base case.



  • For 2 bottles: Taking one rat (R1). If the rat R1 drinks the bottle 1 and dies, then bottle 1 is poisonous. Else the bottle 2 is poisonous. Hence 1 rat is enough to identify
  • For 3 bottles: Taking two rats (R1) and (R2). If the rat R1 drinks the bottle 1 and bottle 3 and dies, then bottle 1 or bottle 3 is poisonous. So the rat R2 drinks the bottle 1 then. If it dies, then the bottle 1 is poisonous, Else the bottle 3 is poisonous.
    Now if the rat R1 do not die after drinking from bottle 1 and bottle 3, then bottle 2 is poisonous.
    Hence 2 rats is enough to identify.
  • For 4 bottles: Taking two rats (R1) and (R2). If the rat R1 drinks the bottle 1 and bottle 3 and dies, then bottle 1 or bottle 3 is poisonous. So the rat R2 drinks the bottle 1 then. If it dies, then the bottle 1 is poisonous, Else the bottle 3 is poisonous.
    Now if the rat R1 do not die after drinking from bottle 1 and bottle 3, then bottle 2 or bottle 4 is poisonous. So the rat R1 drinks the bottle 2 then. If it dies, then the bottle 2 is poisonous, Else the bottle 4 is poisonous.
    Hence 2 rats is enough to identify.
  • For N bottles:

    Minimum number of rats required are = ceil(log2 N))

Below is the implementation of the above approach:

C++

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// C++ program to implement
// the above appraoch
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the minimum number of rats
int minRats(int n)
{
    return ceil(log2(n));
}
  
// Driver Code
int main()
{
    // Number of bottles
    int n = 1025;
  
    cout << "Minimum " << minRats(n)
         << " rat(s) are required"
         << endl;
  
    return 0;
}

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Java

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// Java program to implement 
// the above appraoch
class GFG
{
    public static double log2(int x)
    {
        return (Math.log(x) / Math.log(2));
    }
  
    // Function to find the minimum number of rats 
    static int minRats(int n) 
    
        return (int)(Math.floor(log2(n)) + 1); 
    
      
    // Driver Code 
    public static void main (String[] args) 
    
        // Number of bottles 
        int n = 1025
      
        System.out.println("Minimum " + minRats(n) + 
                           " rat(s) are required"); 
    }    
  
// This code is contributed by AnkitRai01

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Python3

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# Python3 program to implement 
# the above appraoch 
import math
  
# Function to find the 
# minimum number of rats 
def minRats(n):
      
    return math.ceil(math.log2(n)); 
  
# Driver Code 
  
# Number of bottles 
n = 1025
print("Minimum ", end = "")
print(minRats(n), end = " ")
print("rat(s) are required")
  
# This code is contributed 
# by divyamohan123

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C#

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// C# program to implement 
// the above appraoch
using System;
  
class GFG
{
    public static double log2(int x)
    {
        return (Math.Log(x) / Math.Log(2));
    }
  
    // Function to find the minimum number of rats 
    static int minRats(int n) 
    
        return (int)(Math.Floor(log2(n)) + 1); 
    
      
    // Driver Code 
    public static void Main (String[] args) 
    
        // Number of bottles 
        int n = 1025; 
      
        Console.WriteLine("Minimum " + minRats(n) + 
                          " rat(s) are required"); 
    
}
  
// This code is contributed by 29AjayKumar

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Output:

Minimum 11 rat(s) are required


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