Maximum height when coins are arranged in a triangle (original) (raw)

Last Updated : 23 Jul, 2025

We have N coins which need to arrange in form of a triangle, i.e. first row will have 1 coin, second row will have 2 coins and so on, we need to tell maximum height which we can achieve by using these N coins.
Examples:

Input : N = 7
Output : 3
Maximum height will be 3, putting 1, 2 and
then 3 coins. It is not possible to use 1
coin left.

Input : N = 12
Output : 4
Maximum height will be 4, putting 1, 2, 3 and
4 coins, it is not possible to make height as 5,
because that will require 15 coins.

This problem can be solved by finding a relation between height of the triangle and number of coins. Let maximum height is H, then total sum of coin should be less than N,

Sum of coins for height H <= N
H*(H + 1)/2 <= N
HH + H – 2N <= 0
Now by Quadratic formula
(ignoring negative root)

Maximum H can be (-1 + √((1 + 8N)) / 2

Now we just need to find the square root of (1 + 8N) for
which we can use Babylonian method of finding square root

Below code is implemented on above stated concept,

CPP `

// C++ program to find maximum height of arranged // coin triangle #include <bits/stdc++.h> using namespace std;

/* Returns the square root of n. Note that the function / float squareRoot(float n) { / We are using n itself as initial approximation This can definitely be improved */ float x = n; float y = 1;

float e = 0.000001; /* e decides the accuracy level*/
while (x - y > e)
{
    x = (x + y) / 2;
    y = n/x;
}
return x;

}

// Method to find maximum height of arrangement of coins int findMaximumHeight(int N) { // calculating portion inside the square root int n = 1 + 8*N; int maxH = (-1 + squareRoot(n)) / 2; return maxH; }

// Driver code to test above method int main() { int N = 12; cout << findMaximumHeight(N) << endl; return 0; }

Java

// Java program to find maximum height // of arranged coin triangle class GFG {

/* Returns the square root of n. 
Note that the function */
static float squareRoot(float n)
{
    
    /* We are using n itself as 
    initial approximation.This 
    can definitely be improved */
    float x = n;
    float y = 1;
    
    // e decides the accuracy level
    float e = 0.000001f; 
    while (x - y > e)
    {
        x = (x + y) / 2;
        y = n / x;
    }
    
    return x;
}

// Method to find maximum height 
// of arrangement of coins
static int findMaximumHeight(int N)
{
    
    // calculating portion inside 
    // the square root
    int n = 1 + 8*N;
    int maxH = (int)(-1 + squareRoot(n)) / 2;
    
    return maxH;
} 

// Driver code 
public static void main (String[] args)
{
    int N = 12;
    
    System.out.print(findMaximumHeight(N));
}

}

// This code is contributed by Anant Agarwal.

Python3

Python3 program to find

maximum height of arranged

coin triangle

Returns the square root of n.

Note that the function

def squareRoot(n):

# We are using n itself as
    # initial approximation
# This can definitely be improved 
x = n 
y = 1 

e = 0.000001  # e decides the accuracy level 
while (x - y > e):
    x = (x + y) / 2
    y = n/x
    
return x 

Method to find maximum height

of arrangement of coins

def findMaximumHeight(N):

# calculating portion inside the square root
n = 1 + 8*N 
maxH = (-1 + squareRoot(n)) / 2
return int(maxH) 

Driver code to test above method

N = 12 print(findMaximumHeight(N))

This code is contributed by

Smitha Dinesh Semwal

C#

// C# program to find maximum height // of arranged coin triangle using System;

class GFG { /* Returns the square root of n. Note that the function / static float squareRoot(float n) { / We are using n itself as initial approximation.This can definitely be improved */ float x = n; float y = 1;

    // e decides the accuracy level
    float e = 0.000001f;
    while (x - y > e)
    {
        x = (x + y) / 2;
        y = n / x;
    }
    return x;
}

static int findMaximumHeight(int N)
{

    // calculating portion inside
    // the square root
    int n = 1 + 8*N;
    int maxH = (int)(-1 + squareRoot(n)) / 2;

    return maxH;
}

/* program to test above function */
public static void Main()
{
    int N = 12;
    Console.Write(findMaximumHeight(N));
}

}

// This code is contributed by _omg

JavaScript

PHP

`

**Output:

4

**Time complexity: O(log(n))

**Auxiliary space: O(1)