The Lazy Caterer’s Problem
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.
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
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 :
2 4 7 1276
References : oeis.org
This article is contributed by Ashutosh Kumar .If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.