The Lazy Caterer’s Problem

• Difficulty Level : Easy
• Last Updated : 30 Mar, 2021

Given an integer n, denoting the number of cuts that can be made on a pancake, find the maximum number of pieces that can be formed by making n cuts.
Examples :

Input :  n = 1
Output : 2
With 1 cut we can divide the pancake in 2 pieces

Input :  2
Output : 4
With 2 cuts we can divide the pancake in 4 pieces

Input : 3
Output : 7
We can divide the pancake in 7 parts with 3 cuts

Input : 50
Output : 1276 Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Let f(n) denote the maximum number of pieces
that can be obtained by making n cuts.
Trivially,
f(0) = 1
As there'd be only 1 piece without any cut.

Similarly,
f(1) = 2

Proceeding in similar fashion we can deduce
the recursive nature of the function.
The function can be represented recursively as :
f(n) = n + f(n-1)

Hence a simple solution based on the above
formula can run in O(n).

We can optimize above formula.

We now know ,
f(n) = n + f(n-1)

Expanding f(n-1) and so on we have ,
f(n) = n + n-1 + n-2 + ...... + 1 + f(0)

which gives,
f(n) = (n*(n+1))/2 + 1

Hence with this optimization, we can answer all the queries in O(1).
Below is the implementation of above idea :

C++

 // A C++ program to find the solution to // The Lazy Caterer's Problem #include using namespace std;   // This function receives an integer n // and returns the maximum number of // pieces that can be made form pancake // using n cuts int findPieces(int n) {     // Use the formula     return (n * ( n + 1)) / 2 + 1; }   // Driver Code int main() {     cout << findPieces(1) << endl;     cout << findPieces(2) << endl;     cout << findPieces(3) << endl;     cout << findPieces(50) << endl;     return 0; }

Java

 // Java program to find the solution to // The Lazy Caterer's Problem import java.io.*;   class GFG {     // This function returns the maximum     // number of pieces that can be made     //  form pancake using n cuts     static int findPieces(int n)     {         // Use the formula         return (n * (n + 1)) / 2 + 1;     }           // Driver program to test above function     public static void main (String[] args)     {         System.out.println(findPieces(1));         System.out.println(findPieces(2));         System.out.println(findPieces(3));         System.out.println(findPieces(50));     } }   // This code is contributed by Pramod Kumar

Python3

 # A Python 3 program to # find the solution to # The Lazy Caterer's Problem   # This function receives an # integer n and returns the # maximum number of pieces # that can be made form # pancake using n cuts def findPieces( n ):       # Use the formula     return (n * ( n + 1)) // 2 + 1   # Driver Code print(findPieces(1)) print(findPieces(2)) print(findPieces(3)) print(findPieces(50))   # This code is contributed # by ihritik

C#

 // C# program to find the solution // to The Lazy Caterer's Problem using System;   class GFG {     // This function returns the maximum     // number of pieces that can be made     // form pancake using n cuts     static int findPieces(int n)     {         // Use the formula         return (n * (n + 1)) / 2 + 1;     }           // Driver code     public static void Main ()     {         Console.WriteLine(findPieces(1));         Console.WriteLine(findPieces(2));         Console.WriteLine(findPieces(3));         Console.Write(findPieces(50));     } }   // This code is contributed by Nitin Mittal.



Javascript



Output :

2
4
7
1276

References : oeis.org
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