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Total number of different staircase that can made from N boxes
• Difficulty Level : Hard
• Last Updated : 21 May, 2021

Given N boxes of unit dimension, i.e. (1×1 ). The task is to find the total number of different staircases that can be made from those boxes with the following rules:

• The staircase must be in strictly descending order.
• Each staircase contains at least two steps. ( The total steps are equal to the breadth of the staircase.)

Examples

Input : N = 5
Output : 2
The two staircases are the following : Input : N = 6
Output : 3
The three staircases are the following : If we consider total steps = 2, we can observe the fact that the number of staircases is incremented by 1 if N is incremented by 2. We can illustrate the above things from the following image: Now, if the total steps are greater than 2 (assume, total steps = K), then we can take this thing as first create a base (base requires boxes equal to the total steps) for the staircase and put another staircase on it of steps size K and K – 1 have boxes N – K. (because K boxes already used to create base). Thus, we can solve this problem using bottom-up dynamic programming.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the total number of``// different staircase that can made``// from N boxes``#include ``using` `namespace` `std;` `// Function to find the total number of``// different staircase that can made``// from N boxes``int` `countStaircases(``int` `N)``{``    ``// DP table, there are two states.``    ``// First describes the number of boxes``    ``// and second describes the step``    ``int` `memo[N + 5][N + 5];` `    ``// Initialize all the elements of``    ``// the table to zero``    ``for` `(``int` `i = 0; i <= N; i++) {``        ``for` `(``int` `j = 0; j <= N; j++) {``            ``memo[i][j] = 0;``        ``}``    ``}` `    ``// Base case``    ``memo = memo = 1;` `    ``for` `(``int` `i = 5; i <= N; i++) {``        ``for` `(``int` `j = 2; j <= i; j++) {` `            ``// When step is equal to 2``            ``if` `(j == 2) {``                ``memo[i][j] = memo[i - j][j] + 1;``            ``}` `            ``// When step is greater than 2``            ``else` `{``                ``memo[i][j] = memo[i - j][j] +``                             ``memo[i - j][j - 1];``            ``}``        ``}``    ``}` `    ``// Count the total staircase``    ``// from all the steps``    ``int` `answer = 0;``    ``for` `(``int` `i = 1; i <= N; i++)``        ``answer = answer + memo[N][i];   ` `    ``return` `answer;``}` `// Driver Code``int` `main()``{``    ``int` `N = 7;` `    ``cout << countStaircases(N);` `    ``return` `0;``}`

## Java

 `// Java program to find the total number of``// different staircase that can made``// from N boxes` `import` `java.util.*;` `class` `GFG``{``        ``// Function to find the total number of``        ``// different staircase that can made``        ``// from N boxes``        ``static` `int` `countStaircases(``int` `N)``        ``{``            ``// DP table, there are two states.``            ``// First describes the number of boxes``            ``// and second describes the step``            ``int` `[][] memo=``new` `int``[N + ``5``][N + ``5``];``        ` `            ``// Initialize all the elements of``            ``// the table to zero``            ``for` `(``int` `i = ``0``; i <= N; i++) {``                ``for` `(``int` `j = ``0``; j <= N; j++) {``                    ``memo[i][j] = ``0``;``                ``}``            ``}``        ` `            ``// Base case``            ``memo[``3``][``2``] = memo[``4``][``2``] = ``1``;``        ` `            ``for` `(``int` `i = ``5``; i <= N; i++) {``                ``for` `(``int` `j = ``2``; j <= i; j++) {``        ` `                    ``// When step is equal to 2``                    ``if` `(j == ``2``) {``                        ``memo[i][j] = memo[i - j][j] + ``1``;``                    ``}``        ` `                    ``// When step is greater than 2``                    ``else` `{``                        ``memo[i][j] = memo[i - j][j] +``                                    ``memo[i - j][j - ``1``];``                    ``}``                ``}``            ``}``        ` `            ``// Count the total staircase``            ``// from all the steps``            ``int` `answer = ``0``;``            ``for` `(``int` `i = ``1``; i <= N; i++)``                ``answer = answer + memo[N][i];``        ` `            ``return` `answer;``        ``}``        ` `        ``// Driver Code``        ``public` `static` `void` `main(String [] args)``        ``{``            ``int` `N = ``7``;``        ` `            ``System.out.println(countStaircases(N));``        ` `            ` `        ``}` `}` `// This code is contributed``// by ihritik`

## Python 3

 `# Python 3 program to find the total``# number of different staircase that``# can made from N boxes` `# Function to find the total number``# of different staircase that can``# made from N boxes``def` `countStaircases(N):` `    ``# DP table, there are two states.``    ``# First describes the number of boxes``    ``# and second describes the step``    ``memo ``=` `[[``0` `for` `x ``in` `range``(N ``+` `5``)]``               ``for` `y ``in` `range``(N ``+` `5``)]` `    ``# Initialize all the elements of``    ``# the table to zero``    ``for` `i ``in` `range``(N ``+` `1``):``        ``for` `j ``in` `range` `(N ``+` `1``):``            ``memo[i][j] ``=` `0``        ` `    ``# Base case``    ``memo[``3``][``2``] ``=` `memo[``4``][``2``] ``=` `1` `    ``for` `i ``in` `range` `(``5``, N ``+` `1``) :``        ``for` `j ``in` `range` `(``2``, i ``+` `1``) :` `            ``# When step is equal to 2``            ``if` `(j ``=``=` `2``) :``                ``memo[i][j] ``=` `memo[i ``-` `j][j] ``+` `1``            ` `            ``# When step is greater than 2``            ``else` `:``                ``memo[i][j] ``=` `(memo[i ``-` `j][j] ``+``                              ``memo[i ``-` `j][j ``-` `1``])``    ` `    ``# Count the total staircase``    ``# from all the steps``    ``answer ``=` `0``    ``for` `i ``in` `range` `(``1``, N ``+` `1``):``        ``answer ``=` `answer ``+` `memo[N][i]` `    ``return` `answer` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``N ``=` `7` `    ``print` `(countStaircases(N))` `# This code is contributed``# by ChitraNayal`

## C#

 `// C# program to find the total number``// of different staircase that can made``// from N boxes``using` `System;` `class` `GFG``{``    ` `// Function to find the total number``// of different staircase that can``// made from N boxes``static` `int` `countStaircases(``int` `N)``{``    ``// DP table, there are two states.``    ``// First describes the number of boxes``    ``// and second describes the step``    ``int` `[,] memo = ``new` `int``[N + 5, N + 5];` `    ``// Initialize all the elements``    ``// of the table to zero``    ``for` `(``int` `i = 0; i <= N; i++)``    ``{``        ``for` `(``int` `j = 0; j <= N; j++)``        ``{``            ``memo[i, j] = 0;``        ``}``    ``}` `    ``// Base case``    ``memo[3, 2] = memo[4, 2] = 1;` `    ``for` `(``int` `i = 5; i <= N; i++)``    ``{``        ``for` `(``int` `j = 2; j <= i; j++)``        ``{` `            ``// When step is equal to 2``            ``if` `(j == 2)``            ``{``                ``memo[i, j] = memo[i - j, j] + 1;``            ``}` `            ``// When step is greater than 2``            ``else``            ``{``                ``memo[i, j] = memo[i - j, j] +``                             ``memo[i - j, j - 1];``            ``}``        ``}``    ``}` `    ``// Count the total staircase``    ``// from all the steps``    ``int` `answer = 0;``    ``for` `(``int` `i = 1; i <= N; i++)``        ``answer = answer + memo[N, i];` `    ``return` `answer;``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `N = 7;` `    ``Console.WriteLine(countStaircases(N));``}``}` `// This code is contributed``// by Subhadeep`

## PHP

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## Javascript

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Output:
`4`

Time Complexity: O( ).

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