Eggs dropping puzzle (Binomial Coefficient and Binary Search Solution)

Given n eggs and k floors, find the minimum number of trials needed in worst case to find the floor below which all floors are safe. A floor is safe if dropping an egg from it does not break the egg. Please see n eggs and k floors. for complete statements

Input : n = 2, k = 10
Output : 4
We first try from 4-th floor. Two cases arise,
(1) If egg breaks, we have one egg left so we
    need three more trials.
(2) If egg does not break, we try next from 7-th
    floor. Again two cases arise.
We can notice that if we choose 4th floor as first
floor, 7-th as next floor and 9 as next of next floor,
we never exceed more than 4 trials.

Input : n = 2. k = 100
Output : 14

We have discussed the problem for 2 eggs and k floors. We have also discussed a dynamic programming solution to find the solution. The dynamic programming solution is based on below recursive nature of the problem. Let us look at the discussed recursive formula from a different perspective.

How many floors we can cover with x trials?
When we drop an egg, two cases arise.

  1. If egg breaks, then we are left with x-1 trials and n-1 eggs.
  2. If egg does not break, then we are left with x-1 trials and n eggs
Let maxFloors(x, n) be the maximum number of floors 
that we can cover with x trials and n eggs. From above 
two cases, we can write.

maxFloors(x, n) = maxFloors(x-1, n-1) + maxFloors(x-1, n) + 1
For all x >= 1 and n >= 1

Base cases : 
We can't cover any floor with 0 trials or 0 eggs
maxFloors(0, n) = 0
maxFloors(x, 0) = 0

Since we need to cover k floors, 
maxFloors(x, n) >= k           ----------(1)

The above recurrence simplifies to following,
Refer this for proof.

maxFloors(x, n) = ∑xCi
                  1 <= i <= n   ----------(2)
Here C represents Binomial Coefficient.

From above two equations, we can say.
&Sum;xCj  >= k
1 <= i <= n
Basically we need to find minimum value of x
that satisfies above inequality. We can find
such x using Binary Search.

C++

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// C++ program to find minimum 
// number of trials in worst case. 
#include <bits/stdc++.h> 
  
using namespace std; 
  
// Find sum of binomial coefficients xCi 
// (where i varies from 1 to n). 
int binomialCoeff(int x, int n) 
    int sum = 0, term = 1; 
    for (int i = 1; i <= n; ++i) { 
        term *= x - i + 1; 
        term /= i; 
        sum += term; 
    
    return sum; 
  
// Do binary search to find minimum 
// number of trials in worst case. 
int minTrials(int n, int k) 
    // Initialize low and high as 1st 
    // and last floors 
    int low = 1, high = k; 
  
    // Do binary search, for every mid, 
    // find sum of binomial coefficients and 
    // check if the sum is greater than k or not. 
    while (low < high) { 
        int mid = (low + high) / 2; 
        if (binomialCoeff(mid, n) < k) 
            low = mid + 1; 
        else
            high = mid; 
    
  
    return low; 
  
/* Driver program to test above function*/
int main() 
    cout << minTrials(2, 10); 
    return 0; 

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Java

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// Java program to find minimum 
// number of trials in worst case.
class Geeks {
  
// Find sum of binomial coefficients xCi
// (where i varies from 1 to n). If the sum
// becomes more than K
static int binomialCoeff(int x, int n, int k)
{
    int sum = 0, term = 1;
    for (int i = 1; i <= n && sum < k; ++i) {
        term *= x - i + 1;
        term /= i;
        sum += term;
    }
    return sum;
}
  
// Do binary search to find minimum 
// number of trials in worst case.
static int minTrials(int n, int k)
{
    // Initialize low and high as 1st 
    //and last floors
    int low = 1, high = k;
  
    // Do binary search, for every mid, 
    // find sum of binomial coefficients and 
    // check if the sum is greater than k or not.
    while (low < high) {
        int mid = (low + high) / 2;
        if (binomialCoeff(mid, n, k) < k)
            low = mid + 1;
        else
            high = mid;
    }
  
    return low;
}
  
/* Driver program to test above function*/
public static void main(String args[])
{
    System.out.println(minTrials(2, 10));
}
}
  
// This code is contributed by ankita_saini

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Python3

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# Python3 program to find minimum 
# number of trials in worst case.
  
# Find sum of binomial coefficients 
# xCi (where i varies from 1 to n). 
# If the sum becomes more than K
def binomialCoeff(x, n, k):
  
    sum = 0;
    term = 1;
    i = 1;
    while(i <= n and sum < k): 
        term *= x - i + 1;
        term /= i;
        sum += term;
        i += 1;
    return sum;
  
# Do binary search to find minimum 
# number of trials in worst case.
def minTrials(n, k):
  
    # Initialize low and high as 
    # 1st and last floors
    low = 1;
    high = k;
  
    # Do binary search, for every 
    # mid, find sum of binomial 
    # coefficients and check if 
    # the sum is greater than k or not.
    while (low < high):
  
        mid = (low + high) / 2;
        if (binomialCoeff(mid, n, k) < k):
            low = mid + 1;
        else:
            high = mid;
  
    return int(low);
  
# Driver Code
print(minTrials(2, 10));
  
# This code is contributed 
# by mits

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C#

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// C# program to find minimum 
// number of trials in worst case.
using System;
  
class GFG 
{
  
// Find sum of binomial coefficients
// xCi (where i varies from 1 to n). 
// If the sum becomes more than K
static int binomialCoeff(int x, 
                         int n, int k)
{
    int sum = 0, term = 1;
    for (int i = 1; 
             i <= n && sum < k; ++i) 
    {
        term *= x - i + 1;
        term /= i;
        sum += term;
    }
    return sum;
}
  
// Do binary search to find minimum 
// number of trials in worst case.
static int minTrials(int n, int k)
{
    // Initialize low and high
    // as 1st and last floors
    int low = 1, high = k;
  
    // Do binary search, for every 
    // mid, find sum of binomial 
    // coefficients and check if the
    // sum is greater than k or not.
    while (low < high) 
    {
        int mid = (low + high) / 2;
        if (binomialCoeff(mid, n, k) < k)
            low = mid + 1;
        else
            high = mid;
    }
  
    return low;
}
  
// Driver Code
public static void Main()
{
    Console.WriteLine(minTrials(2, 10));
}
}
  
// This code is contributed
// by Akanksha Rai(Abby_akku)

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PHP

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<?php
// PHP program to find minimum 
// number of trials in worst case.
  
// Find sum of binomial coefficients 
// xCi (where i varies from 1 to n). 
// If the sum becomes more than K
function binomialCoeff($x, $n, $k)
{
    $sum = 0; $term = 1;
    for ($i = 1; $i <= $n && 
         $sum < $k; ++$i
    {
        $term *= $x - $i + 1;
        $term /= $i;
        $sum += $term;
    }
    return $sum;
}
  
// Do binary search to find minimum 
// number of trials in worst case.
function minTrials($n, $k)
{
    // Initialize low and high as 
    // 1st and last floors
    $low = 1; $high = $k;
  
    // Do binary search, for every 
    // mid, find sum of binomial 
    // coefficients and check if 
    // the sum is greater than k or not.
    while ($low < $high
    {
        $mid = ($low + $high) / 2;
        if (binomialCoeff($mid, $n, $k) < $k)
            $low = $mid + 1;
        else
            $high = $mid;
    }
  
    return (int)$low;
}
  
// Driver Code
echo minTrials(2, 10);
  
// This code is contributed 
// by Akanksha Rai(Abby_akku)
?>

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Output:

4

Time Complexity : O(n Log k)

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