Find the number of stair steps

Given the total number of bricks T, find the number of stair steps that can be formed by using given bricks such that if step S has bricks B, then step S+1 should have exactly B+1 bricks and total number of bricks used should be less than or equal to number of bricks available.
Note: Number of bricks required to make step 1 of stair is 2, i.e step S = 1 must have exactly B = 2 bricks.

Examples:

Input  : 15
Output : 4
Bricks should be arranged in this pattern to solve for T = 15:



Explanation:
Number of bricks at step increases by one.
At Step 1, Number of bricks = 2, Total = 2
At step 2, Number of bricks = 3, Total = 5
At step 3, Number of bricks = 4, Total = 9
At step 4, Number of bricks = 5, Total = 14

If we add 6 more bricks to form new step, 
then the total number of bricks available will surpass. 
Hence, number of steps that can be formed are 4 and
number of bricks used are 14 and we are left with 
1 brick which is useless.

Input  : 40
Output : 7
Bricks should be arranged in this pattern to solve for T = 40:


Explanation:
At Step 1, Number of bricks = 2, Total = 2
At step 2, Number of bricks = 3, Total = 5
At step 3, Number of bricks = 4, Total = 9
At step 4, Number of bricks = 5, Total = 14
At step 5, Number of bricks = 6, Total = 20
At step 6, Number of bricks = 7, Total = 27
At step 7, Number of bricks = 8, Total = 35

If we add 9 more bricks to form new step,
then the total number of bricks available will surpass.
Hence, number of steps that can be formed are 7 and 
number of bricks used are 35 and we are left with 
5 bricks which are useless.

Approach:
We are interested in the number of steps and we know that each step Si uses exactly Bi number of bricks. We can represent this problem as an equation:
n * (n + 1) / 2 = T (For Natural number series starting from 1, 2, 3, 4, 5 …)
n * (n + 1) = 2 * T
n-1 will represent our final solution because our series in problem starts from 2, 3, 4, 5…

Now, we just have to solve this equation and for that we can exploit binary search to find the solution to this equation. Lower and Higher bounds of binary search are 1 and T.

Below is the implementation for the above approach:

C++

// CPP program to find the number of steps
#include <bits/stdc++.h>
using namespace std;

// Modified Binary search function
// to solve the equation
int solve(int low, int high, int T)
{
    while (low <= high)
    {
        int mid = (low + high) / 2;

        // if mid is solution to equation
        if ((mid * (mid + 1)) == T) return mid;

        // if our solution to equation 
        // lies between mid and mid-1
        if (mid > 0 && (mid * (mid + 1)) > T &&
                (mid * (mid - 1)) <= T )
            return mid - 1;

        // if solution to equation is
        // greater than mid
        if ((mid * (mid + 1)) > T)
            high = mid - 1;

        // if solution to equation is less
        // than mid
        else
            low = mid + 1;
    }
    return -1;
}

// driver function
int main()
{
    int T = 15;

    // call binary search method to
    // solve for limits 1 to T
    int ans = solve(1, T, 2 * T);

    // Because our pattern starts from 2,3,4,5...
    // so, we subtract 1 from ans
    if (ans != -1)
        ans--;

    cout << "Number of stair steps = "
         << ans << endl;
    return 0;
}

Java

// Java program to find the number of steps
import java.util.*;
import java.lang.*;

public class GfG{
    
    // Modified Binary search function
    // to solve the equation
    public static int solve(int low, int high, int T)
    {
        while (low <= high)
        {
            int mid = (low + high) / 2;

            // if mid is solution to equation
            if ((mid * (mid + 1)) == T) return mid;

            // if our solution to equation 
            // lies between mid and mid-1
            if (mid > 0 && (mid * (mid + 1)) > T &&
                (mid * (mid - 1)) <= T )
            return mid - 1;

            // if solution to equation is
            // greater than mid
            if ((mid * (mid + 1)) > T)
                high = mid - 1;

            // if solution to equation is less
            // than mid
            else
                low = mid + 1;
            }
        return -1;
    }

    // driver function
    public static void main(String argc[])
    {
        int T = 15;

        // call binary search method to
        // solve for limits 1 to T
        int ans = solve(1, T, 2 * T);

        // Because our pattern starts from 2,3,4,5...
        // so, we subtract 1 from ans
        if (ans != -1)
            ans--;

        System.out.println("Number of stair steps = " +
            ans);
        }
}

/* This code is Contributed by Sagar Shukla */

Python3

# Python3 code to find the number of steps

# Modified Binary search function
# to solve the equation
def solve( low , high , T ):
    
    while low <= high:
        mid = int((low + high) / 2)
        
        # if mid is solution to equation
        if (mid * (mid + 1)) == T:
            return mid
        
        # if our solution to equation 
        # lies between mid and mid-1
        if (mid > 0 and (mid * (mid + 1)) > T 
                and (mid * (mid - 1)) <= T) :
            return mid - 1
                    
        # if solution to equation is
        # greater than mid
        if (mid * (mid + 1)) > T:
            high = mid - 1;
            
        # if solution to equation is
        # less than mid
        else:
            low = mid + 1
    return -1
    
# driver code
T = 15

# call binary search method to
# solve for limits 1 to T
ans = solve(1, T, 2 * T)

# Because our pattern starts from 2,3,4,5...
#so, we subtract 1 from ans
if ans != -1:
    ans-=1

print("Number of stair steps = ",ans)

# This code is contributed by "Sharad_Bhardwaj".


Output:

Number of stair steps = 4

This article is contributed by Abhishek rajput.

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